Factual error: During the climax of the film, Flight Director Paul Wick is relieved on the spot for failing to account for weather. In reality, NASA would never simply fire the Flight Director on the spot, but rather declare a "contingency" which puts in place a set of procedures used for emergencies.
Factual error: During the G-force tolerance sequence, the testing apparatus involves an astronaut sitting in an exposed chair which spins at high speeds. In actuality, an exposed seat would be extremely dangerous for an astronaut, as even a small bit of debris in the air (such as dust, or something falling from the observation platform) could impact an astronaut with extreme force. NASA's real g-force testing apparatus has an encapsulated cockpit.