In 1947, a group of determined men gathered at a remote Air Force base in the high desert of California. Their goal was to break the sound barrier by using a small rocket-powered test plane called the X-1. The only problem was that others had tried before, and not all had survived. Some thought of the sound barrier as a "demon that lived in the sky", waiting to destroy any who dared confront it.
Chuck Yeager (Sam Shepard), a young wartime ace with a wild reputation, volunteers to fly the X-1, and spends the night before his flight at the local bar. After a few drinks, he and his wife go galloping off on their horses through the twilight, but a tree branch catches Yeager across the chest, knocking him off the horse and cracking a few ribs. Hiding his injury from the flight crew the next morning, Yeager climbs painfully into the plane and prepares to confront the sound barrier "demon".
The little X-1 is carried aloft by a large B-29 bomber, then at the right altitude, is dropped free. Yeager ignites the rocket engine and quickly zooms up into the sky, going faster and faster, closing in on the deadly Mach-1 mark. The plane begins to vibrate, then shake, with Yeager fighting the controls. Suddenly, there is a strange boom, heard by the crew waiting on the ground, and everyone fears that the sound barrier has claimed another life. A moment later, they are surprised to hear Yeager's calm voice crackle over the radio, "Make a note here would you?" This Mach-meter must be busted. It's jumped clear off the scale." Amazed, then jubilant, they realize that Yeager has done it: The sound barrier had been broken. A reporter rushes to the nearest pay phone to report the event and is stopped by an Air Force officer who tells him the nature of the mission is top secret - they don't want the news to reach the Soviet Union.
The desert airbase, now called Edwards, quickly becomes the center for all test and experimental aircraft in the U.S., and every hot-shot jet-jockey looking for a chance to become famous gravitates there. A few years pass, and one such pilot, Gordo Cooper (Dennis Quaid) can't wait to take a crack at becoming "the best pilot anyone ever saw." He finds Edwards already packed with others like himself, with Yeager still at the top of the pyramid.
Unforeseen events halfway around the world change everything for these test pilots. The Communists launch Sputnik, history's first orbiting satellite, and suddenly everything is focused on the race for space. Looking for the first American astronauts, representatives from the newly formed NASA visit Edwards. Cooper sees a chance to stand out from the crowd and volunteers, along with his friends Gus Grissom (Fred Ward) and Deke Slayton (Scott Paulin). Other pilots, such as Alan Shepard (Scott Glenn) from the Navy, and John Glenn (Ed Harris) a Marine flyer, also answer the call.
Tested to the limits of discomfort, pain and exhaustion, the astronaut candidates are slowly weeded out, leaving a mere seven pilots as the handpicked group that will lead America into space. Basking in their glory, they are stunned to hear that the Soviets have beaten them again, launching the first human, Yuri Gagarin, into space. Chagrined, they must now play catch-up with the Russians, and Alan Shepard is chosen to be the first American astronaut.
Before they can launch a human into space, NASA sends a chimpanzee as a test subject. The chimp returns to Earth safely but the astronauts are dismayed that an animal went first. They are further dismayed when a series of test rockets fail to launch, either collapsing on the launch pad or explode in mid-air. The astronauts also quarrel with Wernher von Braun and the other NASA engineers about the design of the space capsule ("space craft") and whether or not it will have windows and a hatch with emergency bolts to escape if anything goes wrong. The scientists initially refuse to listen to the astronauts demands but relent when the men mention that funding can easily run out for space missions and that they'll spill everything to the press, who are kept waiting outside the hangar while the capsule is inspected.
Shepard is strapped into his capsule early on the morning of the launch, he waits patiently while ground control works its way through an endless series of glitches. Shepard's flight was supposed to be a short 15-minute loft into space, and he's been waiting on the pad for hours. Feeling the call of nature, Shepard is forced to "do it in the suit", much to the embarrassment of ground control. However, once he's relieved himself, he demands that they get on with it and fire the rocket. "I'm cooler than you are, so let's light this candle!" Everyone holds their breath, and the button is pushed. Shepard's rocket ignites, and quickly climbs into the sky. Subjected to tremendous stress during the launch and then the fall back to Earth, he survives the flight and is picked up by the waiting helicopter. America has its first astronaut.
Gus Grissom is next and his flight goes well, but during the recovery, the hatch on his capsule is unexpectedly blown off, and when the sea floods in, Grissom nearly drowns and the capsule is lost at the bottom of the Pacific. No one believes his claim that there was a fault in the system, and he's denied the hero's welcome afforded Shepard. The Russians make yet another bold move, placing a second man into orbit while the Americans struggle with short sub-orbital flights. Decisive action is needed, and John Glenn is chosen for the next American mission.
An unreliable rocket, the Mercury Atlas, is chosen to launch Glenn into orbit, and the country watches as the risky flight progresses. Thankfully, Glenn's capsule makes it into orbit, and everyone draws a sigh of relief. Soon however, trouble develops, and there is a serious doubt that the capsule's heat shield will protect Glenn during re-entry. Without it, he'll be incinerated. Facing the fact that there's nothing he can do but try, Glenn fires his retros and begins to fall back through the atmosphere at 18,000 miles per hour. As the heat builds up around the capsule, his radio link to the ground is blanked out, and all they can do is wait. Minutes tick by as Glenn tears through the super-heated air. Amazingly, he survives, and American can finally claim they have at last matched the Russians in the space race.
Back at Edwards, events have bypassed Chuck Yeager. No one cares about high-altitude flight or Mach-speed records now; everyone is talking about spacemen. In one more record-setting attempt, Yeager takes a specially modified F-104 Starfighter up in a dangerous, high-altitude flight. Zooming through the stratosphere once more, he pushes his plane to the limit, climbing higher and higher, pushed to a record altitude by the rocket in the tail. The sky around him grows dark as he approaches the edge of space.
Suddenly the engine begins to stall. The air is too thin and the jet can't keep running at this height. The plane slows as alarm lights flash on the control panel. Yeager looks out at the dark sky around him, and for a moment he can see the stars twinkling just out of reach; he is almost there. But the plane lurches, and begins to fall. There's no control, no power, no way to recover. Yeager is in a deadly flat spin. Spiraling down faster and faster, Yeager struggles to get his craft under control, but as he nears the ground, he must either eject or crash. Pulling the ejection control, his seat is fired out of the doomed plane, but the small rocket in the ejection seat has ignited his flight suit, and he plummets through the clouds, trailing smoke as the flames burn up into his helmet.
On the ground, the crash truck lumbers out over the flat desert toward the crashed plane. The recovery crew fully expects to find Yeager's smashed body in the wreckage, but something off in the distance catches their eye, and they turn toward it. As they draw closer, the shape becomes Yeager, his face badly burned, calmly and proudly walking towards them. He has survived once again.
Intercut with Yeager's flight are scenes of the Mercury astronauts attending a huge Texas banquet thrown by Lyndon B. Johnson. The men watch a performance by fan dancer Sally Rand, and look at each other, knowing they're heroes to all Americans. Gordo Cooper remarks that he's achieved success as an astronaut having a good home, money, and a fine meal, and that he hasn't even been on a space mission yet. When asked who the greatest pilot he ever knew was, Gordo hesitates for a time, perhaps wanting to mention Chuck Yeager, and instead answers "You're lookin' at him."
As the last of the Mercury-7 astronauts prepares for his flight into space, the ground controllers hear snoring over the microphone in the capsule. Gordo Cooper is finally getting his chance to prove he has "the right stuff", but first they have to wake him up. Laughing, they continue the countdown, and Cooper rides the rocket into space, setting records for the longest space flight to date, and proving, if only for a short time, that he is "the greatest pilot anyone ever saw." The narrator tells us that Cooper was the last astronaut to go into space alone.