Not only has Mr. Musk’s company built and flown the world’s largest and most powerful rocket, it has also demonstrated a key technology needed to make the vehicle completely reusable and able to fly again and again quickly, more like a jetliner than a rocket.

Bill Nelson, the NASA administrator, congratulated SpaceX on Sunday’s flight in a post on the social media platform X.

“As we prepare to go back to the Moon under Artemis, continued testing will prepare us for the bold missions that lie ahead — including to the South Pole region of the moon and then on to Mars,” he said.

At 8:25 a.m. Eastern time, the rocket lifted off from the company’s launch site near Brownsville, Texas. Together with the booster called the Super Heavy, the Starship is taller than the Statue of Liberty and its pedestal and will be capable of carrying more than 100 metric tons to orbit.

After the Super Heavy booster, the rocket’s first stage, successfully pushed Starship upward through the densest part of the atmosphere, it dropped away as the upper-stage spacecraft, continued to head toward space.

The booster then ignited some of its 33 engines, looking like a giant cigarette falling at an angle, with the glow of the engines at the bottom. Flames rose along one side of the booster, but that did not seem to knock it off course.

As the booster descended close to the launch tower, the arms closed gently around the booster, catching it before it hit the ground.

When the engines switched off, the booster was still hanging in midair.

A small fire continued to burn at the base of the booster, but the flames soon extinguished.

On the previous Starship test flight, the booster performed similar maneuvers, but over the Gulf of Mexico.

The return of the booster to the launch site was similar to how the first stage of SpaceX’s Falcon 9 rocket re-enters the atmosphere before landing on a floating platform or a landing pad. SpaceX now does that routinely, but the boosters still have to be brought back to the launch site.

Mr. Musk envisions that Starships will be able to fly much more often, so SpaceX built a tower with mechanical arms nicknamed chopsticks. SpaceX also plans to use those arms to catch Starships when they return from orbit.

As the launch tower caught the booster in Texas, the upper-stage Starship spacecraft continued on an arc into space toward the Indian Ocean, essentially the same trajectory as the last flight. For this flight, SpaceX improved the thermal protection on Starship so that it could better withstand the searing heat of re-entry. On the previous flight, the flaps that control the descent almost burned up and disintegrated.

At least one of the flaps looked to suffer damage during re-entry, but Starship survived intact as it performed a soft landing on the water. It then exploded, which was not unexpected. SpaceX was not planning to recover the spacecraft.

A camera that SpaceX had set up on a buoy in the ocean captured Starship’s final moments, and the footage suggests it landed almost exactly where it was aiming for.

While SpaceX’s commentators remained focused on the mission Sunday, the test flight occurred as Mr. Musk invested his time and money in supporting the presidential campaign of former President Donald J. Trump. He appeared beside Mr. Trump at a rally this month in Butler, Pa., wearing a shirt that said “Occupy Mars,” and has asserted that Mr. Trump is the only candidate who will get humanity to Mars.

Valerie Bates, a manager at the Port Isabel Lighthouse State Historic Site, which is about six miles from the launch site, said the Sunday flight was noticeably stronger, at least to her, than earlier launches once the landing was included.

The sound during liftoff is always long and rumbling, she said, but the vibrations were more intense as the booster returned. She said some neighbors reported items falling off walls.

“This was clearly different,” she said, adding that she was at the lighthouse for the event. “There were two to three really strong booms at the landing that felt more impactful. It was stronger, it was louder.”

Ms. Bates said there was no apparent damage to the lighthouse, where the glass has a temporarily shield to protect it from launch vibrations.

Another resident at a greater distance also said he felt the launch and landing.

“My house shook on liftoff and re-entry, all the way in Harlingen, 40 miles away,” said Justin LeClaire, a wildlife biologist with the Coastal Bend Bays and Estuaries Program, which monitors the SpaceX launches. “It genuinely felt like a minor earthquake, had I not known a rocket was launching.”

In public filings, the F.A.A. had said that the air pressure, noise and sonic booms all would be more intense during Sunday’s test, as there was both a launch and a landing, with rocket engines firing. The air pressure, the F.A.A. said in a report made public Saturday, was going to be nearly strong enough to potentially cause minor damage to older plaster on homes not far from the launch site.

Not all area residents reported more intense disruptions.

Keith Reynolds, who lives four miles from the launch site, said the vibrations and noise were not that different to him than they were during previous launches, and he did not see any damage to his house.

“It was an incredible feat,” he said. “I’m ready for the next one and 100 more.”

Planned post-flight inspections will show if the event caused any damage, and if so, how widespread it was. SpaceX, the F.A.A. said, has insurance to cover the cost of any required structural repairs.

The United Launch Alliance this month successfully launched its new Vulcan rocket to orbit for a second time even though it appeared that the nozzle fell off one of the side boosters.

Two other heavy-lift rockets are preparing to take flight this month and in November.

The first rocket is the Falcon Heavy, another launcher built and operated by SpaceX.

The Falcon 9 is SpaceX’s workhorse reusable booster rocket that routinely flies payloads and people to low-Earth orbit. Falcon Heavy is basically three Falcon 9’s strapped together. The Falcon Heavy can generate five million pounds of thrust at liftoff — far below the 16 million pounds that Starship can produce. But it is more powerful than any other vehicle that regularly launches to space these days.

Falcon Heavy first took flight in 2018, and has launched 10 times overall.

As early as Monday, a Falcon Heavy is scheduled to send Europa Clipper, a large NASA scientific spacecraft, on a trajectory toward Jupiter. When the spacecraft reaches Jupiter, it will begin a series of flybys of Europa, an icy ocean moon that scientists think may be a candidate for habitability.

The second heavy-lift vehicle flying soon is New Glenn, an orbital class rocket by Blue Origin. That company was started by Jeff Bezos, the founder of Amazon.

While Blue Origin was created before SpaceX, the company has yet to send a rocket to orbit. So far, its only flights to space have involved New Shepard, a modest reusable suborbital rocket and capsule that briefly sends tourists and cargo to the edge of space before landing back on Earth.

New Glenn, a 320-foot-tall rocket, aims to change that, adding Blue Origin as a player in spaceflight and a serious competitor against SpaceX. Its reusable booster stage’s seven engines will produce 3.85 million pounds of thrust at liftoff, and the company advertises an ability to carry about 100,000 pounds to low-Earth orbit.

The rocket has never flown before, although the engine it relies on, the BE-4, has successfully used in the boosters of the Vulcan rocket.

The first flight of New Glenn was to send two small orbiters to Mars on a mission for NASA called ESCAPADE in October. But NASA did not think the rocket would be ready in time. The ESCAPADE mission was postponed until next year. But the first New Glenn rocket will instead launch a different payload, one for Blue Origin itself, in November.

It will try to demonstrate partial reuse by landing the New Glenn booster stage on a floating platform in the Atlantic Ocean, essentially the same procedure that SpaceX uses to recover most Falcon 9 boosters.

“We will lead the world in space and reach Mars before the end of my term,” he said most recently during a rally in Reading, Pa., on Thursday.

Mr. Trump has not specified whether he means landing American astronauts, or only spacecraft.

But Mr. Musk — who appeared with the former president wearing an “Occupy Mars” T-shirt at a rally in Butler, Pa., earlier this month — has separately promised to send Starships with astronauts to Mars “in 4 years.” He has also warned that humanity will only make it to Mars under a second Trump presidency.

“We will never reach Mars if Kamala wins,” he said of Vice President Kamala Harris.

Earth and Mars pass relatively close to each other once every 26 months. The next such window to head to Mars will be in late 2026, a time frame during which Mr. Musk said uncrewed Starships would test landing on the red planet.

The next launch window after that is December 2028 through January 2029. Mr. Musk says the first astronauts will set off to Mars then.

Mr. Musk’s timeline is thus possible, at least in terms of orbital dynamics.

But Mr. Musk has a long history of offering unrealistic, overly optimistic schedules for his rocket developments. In 2016, when he first announced his Mars rocket, then called the Interplanetary Transport System, he predicted that the first uncrewed SpaceX missions on Mars would launch in 2022, and that the first astronauts to Mars would be taking off this year.

That has not come to pass.

So far, there have been four test flights of Starship. None has been fully successful, although SpaceX has made progress with each. Even with the achievements of the fifth launch, huge technological hurdles remain, including quick turnarounds between launches and refueling Starships while in orbit. Each Starship headed to Mars would most likely need at least dozen or more additional Starship launches to fill up its tanks with methane and liquid oxygen.

SpaceX will have to master many of the same technologies in order for Starship to serve as the lunar lander for NASA’s Artemis III mission, which aims to take astronauts to the surface of the moon near the South Pole.

The task of sending people to Mars is more plausible if you don’t worry about bringing them back alive.

On the International Space Station, toilets still break, but replacements can be sent up within a month or two.

Imagine life in space on the way to Mars without a working toilet.

The life-support system on the Starship would have to reliably work — scrubbing carbon dioxide from the air, recycling water and performing other tasks to keep it habitable — for more than a year. SpaceX would also need spacesuits usable for walking on Mars, and shelters from the radiation of a solar storm.

If the astronauts successfully landed on Mars, the return trip would require more yet-to-be-proven technologies.

For one, the Starship would have to be refueled with methane and oxygen. An experiment on NASA’s Perseverance rover on Mars showed that it was indeed possible to extract oxygen from the Martian air. In two years, it generated 4.3 ounces of oxygen. Starship would need several million pounds of oxygen for the return trip and at least a million pounds of methane, too.

SpaceX could conceivably send additional Starships with the propellants for the return trip, but that adds complexity.

Then there is the question of who would pay for this. These Mars flights would occur at a time when NASA would be busy with its Artemis moon missions. SpaceX already needs two Starship campaigns, an uncrewed test and then the Artemis II mission, to fulfill its contractual obligations to NASA.

There is so far no apparent political groundswell for public financing of Mr. Musk’s Mars dreams, and no obvious business case that would attract the investment of venture capitalists.

There has been widespread evidence of environmental consequences to the region, as detailed in a New York Times investigation in July.

The primary role of F.A.A. for launches and re-entries is to prevent property damage or injury to a member of the public.

After weeks of pressure by Mr. Musk on the agency to speed up its latest review, approval came Saturday for the flight on Sunday. But some environmental caveats were attached.

The report in The Times examined, in part, damage that a Starship launch in June caused to the fragile migratory bird habitat surrounding the launch site, including destroying eggs in nearby nests.

After the story was published, Mr. Musk made fun of the findings. “To make up for this heinous crime, I will refrain from having omelette for a week,” he wrote in a social media post.

In issuing the new launch license, the F.A.A. said SpaceX had agreed to conduct studies, beginning with this launch, of the plume of pea-sized gravel the rocket sends out into the adjacent state park “to determine the distance of the gravel plume and methods for protecting nests during launch events.”

SpaceX will also use drones with infrared cameras — which can detect animal nesting sites — during launches, and it will deploy environmental engineers before and after the launches to look for any harm to these birds and their nests.

The company may also be asked, if the U.S. Fish and Wildlife Service approves, to install temporary shelters to protect the nests during the launches, the F.A.A. environmental report released on Saturday said.

There have also been other points of conflict between SpaceX and the F.A.A.

In September, the F.A.A. proposed $633,000 in fines against SpaceX for violations during launches of the Falcon 9 rockets last year when SpaceX had submitted proposals for changes in its launch procedures without the agency’s approval.

“Safety drives everything we do at the F.A.A., including a legal responsibility for the safety oversight of companies with commercial space transportation licenses,” Marc Nichols, the agency’s chief counsel, said in a statement that announced the fines. “Failure of a company to comply with the safety requirements will result in consequences.”

Elon Musk called the fines “regulatory overreach.”

“The F.A.A. space division is harassing SpaceX about nonsense that doesn’t affect safety while giving a free pass to Boeing even after NASA concluded that their spacecraft was not safe enough to bring back the astronauts,” Mr. Musk wrote on X. “There need to be resignations from the F.A.A. leadership.”

In testimony at a congressional hearing, officials from other space companies also said the approval process for launching space missions needs streamlining, but they have not criticized the F.A.A. using language as harsh as Mr. Musk’s.

The Super Heavy booster for Starship, however, is much bigger. It is 232 feet tall and nearly 30 feet wide with 33 Raptor engines. The Falcon 9 booster is about 135 feet tall and 12 feet in diameter with nine of SpaceX’s smaller Merlin engines.

And instead of just landing the booster, SpaceX will try to catch it out of the air using what looks like a giant pair of chopsticks that will close around the booster as it arrives at the tower. If that works, the booster will essentially be in position to be refueled for its next launch, saving the time of transporting it back to the launchpad.

During the last flight, the booster made a successful simulated landing into the Gulf of Mexico. SpaceX said that if anything was not quite right, the booster would dunk into the Gulf of Mexico again. The catch attempt will require a manual command from the flight director, the company said.

The second stage — the Starship spacecraft — will repeat the trajectory of the fourth flight, on a suborbital trajectory that will take it to the Indian Ocean. During the last flight, Starship was able to survive re-entry. But only barely: The steering flaps that guided it through the atmosphere were slowly burning up and disintegrating.

SpaceX has added additional thermal protection to Starship to help keep it intact.

Each of Starship’s four flights has achieved steady progress since the first test launch in April 2023, but they have not been without hiccups.

Flight 1: April 20, 2023

This Starship prototype made it off the launchpad. But, early in the flight, a cascade of engine failures and fires in the booster caused the vehicle to tumble before officials on the ground sent a command to destroy it 24 miles above the Gulf of Mexico.

But even that did not go according to plan — 40 seconds passed between the self-destruct command being issued and the rocket finally exploding. The launch also hurled debris across a wide area and generated clouds of dust, which reached a small town miles away from the launchpad at the southern tip of Texas.

Elon Musk, the founder of SpaceX, said in an audio chat on the social media platform X after the flight that it was “Obviously not a complete success, but still nonetheless successful.”

Flight 2: Nov. 18, 2023

In November, the second Starship launch traveled much farther, with the upper stage traveling for seven minutes and reaching an altitude of about 90 miles and a speed of about 15,000 miles per hour.

That was itself an achievement because the company completed a step called hot-staging, during which the upper-stage engines ignite before the stage detaches from the lower Super Heavy booster.

According to SpaceX’s “fail fast, learn faster” approach toward rocket design, avoiding a repeat of past failures counts as major progress.

Still, both stages of the rocket ended up exploding.

Because the upper spacecraft was empty, extra liquid oxygen was loaded to simulate the weight of a future payload it could carry to orbit. But when the extra oxygen was dumped, a fire started, which set off the flight termination system, destroying the spacecraft.

Earlier in the flight, after the lower and upper stages separated, several of the 33 engines in the Super Heavy booster shut down and then one blew up, causing the vehicle’s destruction over the Gulf of Mexico.

Flight 3: March 4, 2024

The third try turned out to be closer to the charm as the Starship upper stage traveled about halfway around the Earth.

As it coasted in space, the upper Starship stage tested the spacecraft’s payload door, and the company demonstrated an ability to move propellant between two tanks inside the vehicle.

But while coasting at the highest point of its trajectory, Starship began rolling out of control. Cameras on board captured the orange glow of hot plasma beneath the spacecraft. Some 49 minutes after launching, it disintegrated.

Earlier in the flight, the Super Heavy booster was to simulate a landing over the Gulf of Mexico. But six of 13 engines used for that maneuver shut down early.

Bill Nelson, the administrator of NASA, congratulated SpaceX on what he called a “successful test flight” of the system.

Flight 4: June 6, 2024

The fourth flight accomplished an ambitious set of goals for SpaceX.

After the Super Heavy booster dropped away from the upper stage, the booster was able to gently set down in the Gulf of Mexico. The second-stage spacecraft then traveled halfway around the world, surviving the searing temperatures of re-entering the atmosphere while making a controlled splashdown in the Indian Ocean.

The flight was not flawless, and challenging technical hurdles remain. For instance, heat-resistant tiles were lost from the surface of the spacecraft. And, during atmospheric re-entry of the upper stage, pieces began peeling away from one of the steering flaps near the top of the spacecraft, though the flap continued to function.

The successes, surpassing what was accomplished during the previous test flight in March, offered optimism for the vision of a rocket that is the biggest and most powerful to date, yet entirely reusable.

It has the most engines ever in a rocket booster: The Super Heavy has 33 of SpaceX’s powerful Raptor engines sticking out of its bottom. As those engines lift Starship off the launchpad in South Texas, they will generate 16 million pounds of thrust at full throttle.

An even more transformative feature of Starship is that it is designed to be entirely reusable. The Super Heavy booster is to land much like those for SpaceX’s smaller Falcon 9 rockets, and Starship will be able to return from space belly-flopping through the atmosphere like a sky diver before pivoting to a vertical position for landing.

If SpaceX has regularly pull that off, that means the really expensive pieces — like the 33 Raptor engines in the Super Heavy booster and six additional Raptors in Starship itself — will be used over and over instead of thrown away into the ocean after one flight.

That has the potential to cut the cost of sending payloads into orbit — to less than $10 million to take 100 tons to space, Mr. Musk has predicted.

Starship and Super Heavy are shiny because SpaceX made them out of stainless steel, which is cheaper than other materials like carbon composites. But one side of Starship is coated in black tiles to protect the spacecraft from the extreme heat that it will encounter if it gets far enough in its flight to re-enter the atmosphere.