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, and he is grounded.
His 2 1/2-year-old daughter, Karen, is undergoing treatment for a brain tumor. Desperate to save her, Armstrong keeps a detailed log of her symptoms and feverishly tries to find possible treatments, but she dies soon afterwards. Grief-stricken, Armstrong applies for Project Gemini and is accepted to NASA Astronaut Group 2. Armstrong, his wife Janet, and their son Rick move to Houston alongside other astronaut families. Armstrong 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. Armstrong and Janet have a second son, Mark.
By 1965, the family has settled in Houston, and Armstrong awaits selection for a crew. After the Soviets complete the first extravehicular activity (EVA), Armstrong is informed that he will be the commander of 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, the spacecraft begins to roll at a rapid rate. After nearly blacking out, Armstrong activates the RCS thrusters and safely aborts the mission. Armstrong initially faces criticism, but NASA determines the crew is not at fault. 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, and Armstrong hears the news while representing NASA at the White House. The next year, after ejecting from the Lunar Landing Research Vehicle in an accident that could have killed him, Armstrong is selected to command Apollo 11, and Slayton informs him that it will likely be the first lunar landing. As the mission nears, Neil becomes increasingly preoccupied and emotionally distant from his family. Just prior to the launch, Janet confronts Armstrong about the possibility that he won't survive the flight and insists that he should explain the risks of the mission to their young sons. After telling them about the risks he faces, Armstrong says goodbye to his family and departs.
Apollo 11 successfully launches and arrives at the Moon four days later. Armstrong and Aldrin undock in the Lunar Module and begin the landing attempt. They realize that they are coming in too fast and are going to overshoot their designated landing site. The new landing site turns out to have much rougher terrain, forcing Armstrong to take manual control of the spacecraft. He lands the lunar module successfully with minimal fuel remaining. After setting foot on the Moon, Armstrong drops Karen's bracelet into Little West crater. With their mission complete, the astronauts return home and are placed in quarantine, where the film of John F. Kennedy's 1962 speech "We choose to go to the Moon" is shown, and Armstrong and Janet share a moment of tenderness.