Factual error: The movie takes place in 1961 but the police car escorting the women to work is a 1964 Galaxie 500.
Character mistake: A TV news announcer describes Alan Shepard's first manned launch into space as him being lifted to "an altitude of 116 miles per hour." That's not an altitude, that's a speed. It should have just been "116 miles."
Continuity mistake: When Katherine realises she can use Euler's Method to calculate how to bring John Glenn safely back to earth, she is seen wearing a wristwatch when she's at the chalkboard. The watch is not on her when she leaves to get a book about Euler's Method. It reappears when she's back at the chalkboard.
Revealing mistake: When the ladies are being escorted by the police to NASA, their Chevy is in park.
Factual error: The eight stripe IBM logo is shown on a portfolio. That logo did not exist until 1972.
Factual error: The green car that the 3 ladies are always in has a shot where you can see the brand of tires being a Hankook Optimo. This was in 1961 and these radial tires didn't exist yet.
Factual error: The truck that drives Neil Armstrong to the launch pad is an M939 series 5 ton, which did not go into service until 1982.
Factual error: When Colonel Jackson comes to town to organize a National Guard unit, he is wearing the Army Green military uniform. It only has the branch insignia (infantry) and the US lapel. It has no name tag, no shoulder insignia showing what unit he was assigned to, no shoulder unit crests and no award ribbons. A person with that rank must have been in the Armed Forces at least 16-18 years and no ribbons?
Continuity mistake: When Katherine is working out the math in the conference room for John Glenn's start off to the Bahamas there is no chalk or pencils on the desk, but they reappear when Al Harrison hands Katherine the chalk.
Continuity mistake: When John Glenn is attempting re entry, Dorothy Vaughn is driving out to the beach/country and she stops the car near a barn where there is a dark and creamy two tone car parked. The scene cuts straight back to the city to see the other girls watching re entry and the exact same two tone car is parked on the side of the road.
Continuity mistake: In the scene in the briefing room, some of the calculations on the blackboard change writing styles between shots, such as brackets moving closer to numbers.
Continuity mistake: In the last scene in which Vivian gives Dorothy an assignment, her lanyard is at first hanging neatly from her neck. The camera goes back and forth between her and Dorothy, and when you last see Vivian, the ball chain for the lanyard is no longer hanging neatly; it is pulled to the side because it's caught on her jacket lapel.
Factual error: In at least two places in the film an IBM Selectric is used. They are: (a) A room at the beginning of the movie where typewriters are shown; (b) Near the end when Katherine Johnson (Taraji P. Henson) is typing up her report. The Selectric was introduced on July 31, 1961 (the year most of the movie takes place). It is impossible for it to have been used during the first scene as that takes place prior to Shephard's flight, which was on May 5, 1961. Glenn's flight in Friendship 7 took place on February 2, 1962. So the Selectric may have been used by Johnson at that time, but it is extremely unlikely. IBM had expected to sell 20,000 in the first 6 months and had orders (not deliveries) for 80,000 of them by the end of 1961. They were rare. NASA would not have gotten one, as law offices and similar business were IBM's bread and butter.
Factual error: When John Glenn is in the van being transported to the launch pad, the van is being escorted by a police vehicle with Virginia markings. (The launch pad is in Florida).