Racial segregation was still very much a major problem in the U.S. in 1961, but the specifics in "Hidden Figures" are a bit off. Although the West Area Computing unit of Langley did exist, it was abolished along with all other segregated facilities when the National Advisory Committee for Aeronautics (NACA) became NASA in 1958. At the time the movie is set, NASA and all of its divisions were already integrated.
That didn't mean discrimination no longer existed, of course — especially considering segregation laws were still in place in Virginia at the time. Shetterly's book does mention Mary Jackson's frustration at having to look for a colored bathroom during her time at Langley, and it is true that, up until 1958, the Langley Research Center was heavily segregated despite the significant presence of Black staff. But Katherine Johnson herself never bothered to look for a colored bathroom and just used whichever one was closest. In fact, she has gone on record saying she did not personally feel discriminated against at NASA (via History vs Hollywood).
Still, Mary Jackson did have to file a petition with the City of Hampton to be able to attend classes at the whites-only Hampton High School, and, even though the process was not as taxing as in the movie — she was granted special permission and did not have to go to court — Jackson often vented about her distress at the absurdity of segregation to her mentor Kazimierz Czarnecki, upon whom the character of Karl Zielinski is based.
