The actual working relationship between the engineers and women was not as hostile as it appears in the film. While there were clearly racial issues at play, the majority of the engineers were able to work with the computers with no issues.
The issue with the bathrooms was not something Katherine Johnson experienced, but rather lived by Mary Jackson. In fact, it was this incident that resulted in Jackson ranting to a colleague which got her placed on the wind tunnel team. Katherine Johnson simply refused to use colored restrooms.
When Taraji P. Henson signed on for the lead role, she met with the real-life Katherine Johnson, who was 98 years old, to discuss the character she was about to portray. Henson learned that Johnson had graduated from high school at age 14 and from college at age 18, and was still as lucid as anyone years younger. After the film was screened for Johnson, she expressed her genuine approval of Henson's portrayal, but wondered why anybody would want to make a film about her life.
While John Glenn did specifically request that Katherine Johnson review all of the numbers for the Friendship 7 mission before he would agree to go through with it, he did so weeks before the mission actually took place, not when the countdown to launch was nearing at Cape Canaveral.
One of the ways that Katherine experiences workplace discrimination is when her coworkers require her to use a separate coffee pot. Whenever the office's coffee area is shown, the brand of coffee that they use, Chock Full o'Nuts, is also visible. The use of this brand in the context of segregation is historically relevant. In 1957, Chock Full o'Nuts was one of the first major New York corporations to hire a black executive as a corporate vice-president. The man they hired, retired baseball legend Jackie Robinson, made history by being the first person to break the color barrier in professional baseball.
The set where IBM genius Dorothy Vaughn's house was filmed (where the 3 ladies played cards and danced) was actually an historic house in Atlanta where civil rights legends Ralph Abernathy and Martin Luther King Jr. met.
Colors were key to setting the films mood. "Cold" sets at NASA - where calculations took place - were filmed in sterile whites, grays, and silvers; sharply contrasted against the "warm" sets of Kevin Costner's office and the ladies' homes.
On the day that the scene was filmed in which Paul Stafford (Jim Parsons) is speaking to the NASA engineers in the Space Task Group office about needing to develop the math for re-entry, there was an extra face in the crowd. Mark Armstrong, son of Apollo 11 astronaut Neil Armstrong, had been invited by actor Ken Strunk to have a cameo appearance in the scene, and joined the other actors representing the NASA engineers.
In the movie when John Glenn is in his space suit and going to the launchpad, he turns to the man next to him and asks if he has the time. In reality, Glenn was wearing a Heuer 2915A model watch on his left wrist and would have had no need to ask anyone else the time. The watch had two stopwatches in it. Astronauts would have required watches in case they had to fly manual and synchronize their actions with The Capcom.
The movie was part of a recurring gaffe at the Golden Globes when two presenters kept calling the film's title as "Hidden Fences", due to Fences (2016) also being nominated for the same event - in fact, they both competed for the Best Supporting Actress award.
The trivia item below may give away important plot points.
The film accurately portrays what was believed to be a heat shield failure upon re-entry. The film did not provide further explanation, but it turns out a micro-switch on the landing bag, which the heat shield was connected to, was faulty. That caused an indication that the landing bag was deployed and thus the heat shield probably would not have "ablated" properly. The flaming debris seen upon re-entry was not the heat shield but the retrorocket pack, which wasn't jettisoned, in an attempt to hold the landing bag in place. At the time, however, John Glenn had no way of knowing that.