Factual error: In the opening scene the x-plane approaches at a very high speed and abruptly makes a very sharp turn to climb straight up. Even with pressure suits this high-speed maneuver would not only kill the pilots but snap the wings off the plane as well. X-planes were carried up to high altitudes before their engines were ever lit up. Flying an x-plane at ground level just was not done.
Factual error: This line of thinking is repeated twice, once by the NASA scientist and once by the Tommy Lee Jones character: "To get to the moon, you only have to go halfway - gravity will take you the rest of the way". Actually, since Earth's gravity is 6 times greater than the moon's, you would have to go 6/7ths of the way, otherwise you return to Earth.
Continuity mistake: When Frank is stopped in the lobby after leaving Gerson's office if you look out the window behind Frank you see a couple walk by, a woman in a light suit and a man next to her. The shot than shifts to Gerson and back to Frank, you again see the couple walking by, within 30 seconds.
Factual error: A couple of times during the film, someone exclaims in horror that if this-or-that system fails, they will have to land the shuttle dead-stick (unpowered; no chance to veer off and try again). Actually, all shuttle landings are dead-stick; they are computer guided but nevertheless are unpowered, one-shot deals.
Continuity mistake: During the climactic landing sequence at the end, the point of view shifts rapidly from head-on to alongside and back. Watch carefully at the first side-on view just as they're touching down; the nose of the shuttle says "Columbia". When it switches to the next head-on view, the wings say "Daedalus".
Factual error: At one point, the rotating solar power panels on IKON slam into the Shuttle, wrapping around the nose. That's bad for the astronauts! The tiles all over the Shuttle are used to absorb the tremendous heat of reentry. They are also very brittle. An impact like that would have torn them off by the hundreds, making reentry impossible.
Factual error: The flight director talks directly to the crew several times. In order to avoid a lot of confusing chatter, only CapCom can talk to the crew.
Continuity mistake: The character Jerry, played by Donald Sutherland, snaps off one arm of his sunglasses that the doctor gave him. He uses it to assist him in fixing the door that was jammed shut. When they are suited up and coming back to Earth it shows him with the sunglasses back on from both sides of his head and the arm is on the glasses, not broken off anymore.
Other mistake: During the landing, when Jerry is getting Ethan and Roger out of the shuttle using the escape pole, the two people going out (presumably the stuntmen) look to in fact be jumping out the hatch, instead of just being pushed/thrown by Jerry.
Factual error: They only have a month to train. In that time they would train for the mission and just let a trained pilot/commander fly. Also, they wouldn't take time out from training to go to CA to be on Leno.