Continuity mistake: At the end of the movie Armstrong is standing on the surface of the moon and turning towards Earth. At that point the sun is exactly behind him, so Earth should be fully lit. In the next scene Earth (and the moon) are lit from the left, not directly in front as they should be. (02:05:20)
Factual error: The interior shots of the Gemini and Apollo Spacecraft show worn and dirty panels, knobs, switches and circuit breakers. The movie most likely used some original cockpit trainers, but in reality the astronauts were flying brand new spacecraft that were spotless.
Factual error: When they are in quarantine after returning to Earth, they are in a room with several magazines. One of the magazines is an issue of National Geographic with the cover image of them standing on the moon. That is the December 1969 issue of National Geographic, whereas they were actually in quarantine in late July/early August 1969.
Factual error: The Apollo 11 flight shows the sky turning from blue to black through the capsule window, then there is an external shot showing a vapour cone forming on the rocket, which is characteristic of low-altitude transonic flight.
Factual error: The terrain is depicted as riddled with deep craters with sheer walls, and Eagle appears to head straight for the imposing cliffs of a large one (crater West). In fact the slopes of lunar craters are gentle, rarely greater than 35°, and the landing site was selected for its relative smoothness. There were no large hills, high cliffs, or deep craters on the approach that could cause incorrect altitude signals to Eagle's landing radar. The crater Armstrong visited, later dubbed "Little West," is only 60 meters east of the lunar module. It is approximately 33 meters in diameter and only 4 meters deep.
Factual error: Armstrong sees the first quarter moon through one of Columbia's windows during the launch. However, the Command Module was sheathed in a protective shell that included an escape tower in the event that the launch had to be aborted. According to Michael Collins in "Carrying the Fire", the windows were covered until the tower and shroud were jettisoned when the Saturn V reached an altitude of 60 miles. Additionally, on July 16, 1969, the moon was two days past New Moon, only 4% illuminated, and would have been too thin and too near the sun's glare (from Earth's perspective) to be seen without optical aid. It did not reach the phase depicted until July 22. Later the same day, when Columbia docks and pulls Eagle out of the second stage housing, the Earth appears just shy of a first quarter phase. In fact, the Earth was nearly full, since Earth's phases are always opposite those of the moon.
Factual error: When Neil Armstrong drops the bracelet into the crater he lets it drop straight down. Yet the bracelet appears to fall a long way, implying a very sheer cliff, close to 90°. This is not natural for a lunar crater, as the maximum steepness of crater wall is determined by the material's angle of repose (how steep before it avalanches) which has typically been observed to be approximately 45° for lunar regolith.
Factual error: Early in the movie, Neil is looking for additional treatments for Karen's brain cancer. He is looking at a medical journal that shows for only a couple of frames a type of brain image (probably a sagittal section MRI) that would not be available for about 30 years or more.
Factual error: Neil Armstrong actually never trained on the multiple-axis space test inertia facility (MASTIF) gimbal system. All the Mercury astronauts did, but not the Gemini guys. So when his spacecraft spun out of control, Neil brought it back without the experience on the MASTIF simulator.
Factual error: When Armstrong is inspecting the LEM right after docking with it, he uses a flashlight. We then see him turn it off, but a "click" is heard. The flashlights used during the Apollo missions would be turned on/off by rotating the collar, not by a button. It seems like the editors added a click for some effect.
Factual error: When Neil is returning to Earth in the X-15 it shows him flying through clouds and cuts back to an rapidly decreasing altitude reading of around 70 000 feet. Clouds can not form at 70 000 feet because the air is too cold and thin. Earlier in the scene they show Neil supposedly 'flying through the clouds' at 40 000 feet while the X-15 is still under the wing of a B-52. Clouds that thick can't get up to 40 000 feet unless it is a thunderstorm.
Factual error: As depicted in the film, the lunar lander touches down and settles onto the surface of the moon (as shown by the shaking of the landscape stopping). Buzz then reports a "contact light" and Neil orders the engines turned off. In reality, three of the LEM's legs had probes extended for a couple of feet below them, and the contact light would activate when one of the probes touched the surface. This meant the ship was still several feet from touchdown, so the astronauts would turn off the engine and let the moon's gravity complete the landing cycle. So either the film's depiction of the contact light being called would be several seconds too late, or the actual touchdown is depicted as being too early. Take your pick.
Factual error: The Apollo 11 lander Eagle detaches from command module Columbia with its landing legs already extended. Neil is then shown saying "the Eagle has wings" as if to mean the ship is flying on its own. Actually, the ship's legs wouldn't be extended until after the undocking. It was the extension of the legs that prompted Neil's quote about the Eagle having wings.
Factual error: The Apollo 11's stage separation is depicted incorrectly from the onboard view. It shows the interstage falling away, which was done for the last time on Apollo 6, whose footage is often mis-represented as being from Apollo 11, which did not have the onboard film. Also this separation in First Man is the first stage separation, shown by the multiple nozzles of the second stage, but is depicted as being immediately before trans-lunar injection, which would in fact be the second stage separation.