In 1961, NASA test pilot Neil Armstrong is flying the X-15 rocket-powered spaceplane when it inadvertently bounces off the atmosphere. Although he manages to land the plane in the Mojave Desert, his colleagues express concern that his recent record of mishaps is due to distraction.

His 2-year-old daughter, Karen, is undergoing treatment for a brain tumor. Desperate to save her, Neil keeps a detailed log of her symptoms and researches possible treatments, but she dies soon afterward. Grief-stricken and suspecting he has been grounded, Armstrong applies for Project Gemini and is accepted to NASA Astronaut Group 2. With his wife Janet, and their son Rick, Neil moves to Houston alongside other astronaut families. He befriends Elliot See, another civilian test pilot, and Ed White. As Armstrong begins training, Deke Slayton impresses upon the new astronauts the importance of the Gemini program, as the Soviet Union had reached every milestone in the Space Race ahead of the United States. Neil and Janet have a second son, Mark.

In 1965, after the Soviets complete the first extravehicular activity (EVA), Armstrong is informed that he will command Gemini 8, with David Scott as the pilot. Prior to the mission, See and Charles Bassett are killed in a T-38 crash, deepening Armstrong’s grief at the string of recent losses. Armstrong and Scott successfully launch on Gemini 8 and dock with the Agena target vehicle, but soon afterward, a malfunction causes the spacecraft to roll at an increasingly dangerous rate. After nearly blacking out, Armstrong activates the RCS thrusters and safely aborts the mission. He initially faces criticism, but NASA determines the crew is not at fault and the mission is rated a “success”.

Later, White reveals that he has been selected for the Apollo 1 mission, along with Gus Grissom and Roger Chaffee. During a launch rehearsal test on January 27, 1967, a fire kills White and the Apollo 1 crew. Armstrong learns the news while representing NASA at the White House. The next year, after Armstrong ejects from the Lunar Landing Research Vehicle in an accident that could have killed him, Slayton informs Armstrong that he has been selected to command Apollo 11, which will likely attempt the first lunar landing. As the mission nears, Armstrong becomes increasingly preoccupied and emotionally distant from his family. Prior to the launch, Janet confronts Neil about the possibility that he might not survive the flight and insists that he explain the risks of the mission to their young sons. After telling them about the risks he faces, Armstrong says goodbye to his family.

Three days after launch, Apollo 11 enters lunar orbit. Armstrong and Buzz Aldrin undock in the Lunar Module Eagle and begin the landing. The landing site terrain turns out to be much rougher than expected, forcing Armstrong to take manual control of the spacecraft. He lands Eagle successfully at an alternative site with less than 30 seconds of fuel remaining. After setting foot on the Moon, Armstrong utters his famous line, “That’s one small step for man, one giant leap for mankind.” Later he drops Karen’s bracelet into Little West crater. With their mission complete, the astronauts return home and are placed in quarantine, where they watch footage of John F. Kennedy’s 1962 speech “We choose to go to the Moon” on television, and Neil and Janet share a moment of tenderness