The Right Stuff describes the efforts of the U.S. to put a man into space. After WW II, the Air Force tested supersonic rocket planes such as the X-15 and the X-20. The film introduces Chuck Yeager, the legendary test pilot who was the first man to fly faster than the speed of sound. (Yeager had "the right stuff.") In 1958, when the Russians launched the first man-made satellites, NASA's Mercury program was established to send a U.S. astronaut into space as soon as possible. The movie shows the selection, training and flights of the first U.S. astronauts and how they discovered their own tremendous popularity. In one incident they used that popularity to force NASA and its engineers to install manual controls in the spacecraft rather than to rely exclusively on computers to pilot the missions. (This victory saved the lives of several astronauts who had to fly their spaceships when the automatic guidance systems or various pieces of equipment failed.)

The film describes how the mantle of "the right stuff" shifted from USAF test pilots to the Mercury Astronauts. On the way, we are shown the terror that Sputnik and Soviet superiority in space caused the American people, the initial failures and difficulties of the American space program, and the strong camaraderie that developed among the original seven astronauts, despite their intensely competitive natures. The film is historically accurate in all except minor points and is closely based on the book.