The movie is set in 1961 at the height of the Cold War. At Langley Research Centre in Hampton, Virginia, a group of women are employed to use and calculate data for the all-male flight team. This is the story about the jobs of Katherine Goble, Mary Jackson, and Dorothy Vaughan; three African-American women who played vital roles in the success of John Glenn's launch into space and his safe return. Mary Jackson is working in the West Area computers of Langley Research Centre, which is a segregated section dedicated for all African-American female professionals. She was appointed by Mr. Jacka, her supervisor, to help the team of the all-male engineers for the project Atlas in NACA, the predecessor of NASA. Although she lacks the relevant degree to advance her to an engineer, Mary was advised by a Polish-Jewish friend who is working as one of the engineers to attend night classes on accelerated engineering courses. Dorothy Vaughan is the de facto supervisor of the West Computing Centre. She has been successfully recommended by Mrs. Mitchell, her superior, for her transfer from a temporary worker to a full-time job.
Dorothy notices the installation of a USP by IBM made in the laboratory and she is afraid that her colleagues may lose their jobs because of the introduction of a new electronic computer. Therefore, she decides to teach herself FORTRAN, the computer language used by IBM in secret, and she is assisted by a white co-worker, Karl Zielinski, in her studies so that she could find a supervisor post in the programming sector. On the other hand, Katherine Goble is a highly talented mathematician. She is offering her skills in analytical geometry and she is assigned to check the calculations of the engineers in the Space Task Group. After working together with supervisor Paul Stafford, Katherine is introduced to the new IBM as given by Mr. Harrison, the head of the Space Task Group. Harrison has assigned Katherine to check the output from the computer, which means she is the one that should know as much as the engineers do, and she is more important than others because she decides what they get to know. Eventually, John Glenn, the astronaut, was very impressed by the critical calculations done by Katherine and he trusts her work. As we all know, John Glenn makes a successful flight to circle the Earth three times and he returns safely, which is a remarkable milestone in the history of NASA.