Factual error: The shuttle's original mission was to service Hubble, yet when the shuttle is wrecked, Kowalsky moves with Stone to the ISS, which just happens to be "a short hike away." Hubble orbits at an altitude of 350 miles/560km, while the ISS does so at an altitude of about 250 miles/410km. Furthermore, even if they had been able to see the ISS from Hubble's orbit, they would have only seen it speed ahead, as their orbital velocities are very different: 7.66km per second for the ISS and 7.5km per second for Hubble.
Factual error: There are several scenes which depict space debris reaching the astronauts' location, with catastrophic results. While the impact of these collisions are probably realistic, it's highly improbable that this debris would be visible (you can spot many objects approaching, missing or hitting Sandra and George's location) mainly because of their ultra high speed relative to the astronauts' area. Remember, we are talking about orbits with speeds in the order of several 10k's of km/h.
Factual error: When the Chinese space station is de-orbiting, and the atmosphere is stripping parts off the outside, Dr Ryan Stone is inside with objects floating about her. In reality, there would be a small deceleration caused by the atmospheric drag that would pull all objects to the front of the craft.
Continuity mistake: When Ryan mentions her daughter, and Matt turns off his music, he hold up his right arm and looks in the mirror, but the shot of the mirror itself is on his left hand.
Factual error: When Stone makes her transfer from the Soyuz emergency escape system to the Chinese Station, she takes 3+ minutes from fastening her helmet to being inside the station. Her space suit does not include an oxygen supply. Only residual air (from the Soyuz spacecraft) is available to her, and a lot of action occurs. She couldn't do it. Too long a time, too little air. The Sokol spacesuit portrayed is intended for intra vehicular operation and requires external sources of oxygen and ventilation to be functional (as depicted in scenes before).
Continuity mistake: When Sandra reaches the Soyuz space capsule she turns the oxygen 'on' by turning the red knobs counter-clockwise. When she turns the oxygen off she turned the knobs clockwise. But when she turns the oxygen back on she turns the knobs clockwise again.
Factual error: When Stone makes it to the Chinese station, it is experiencing reentry. This is nothing but a drama-making mistake, as the station would have needed to have been actively boosted down/slowed down for it to have been a deliberate deorbit, or have had its navigation completely neglected for months or years for its orbit to decay to that point. In either of these two cases, it could not have been orbiting at a fixed position with the otherwise-stable ISS.
Continuity mistake: After the rope keeping Ryan and Matt tethered together is snapped, Ryan continues drifting downwards towards the Earth. The shot continues panning until the camera stops with the camera high up. As she falls, the ropes from the parachute around the satellite are visibly on Ryan's right. In the next shot from low down, the ropes from the parachute have moved to Ryan's left.
Factual error: Space debris from an exploded satellite orbiting the earth catches up with Ryan every 90 minutes (as she goes from the Explorer to the ISS to the Chinese space station, all roughly at the same altitude). This is physically impossible. To do this, the debris would have to be traveling fast enough to catch up with the orbiting space stations every 90 minutes. However, objects orbiting at different speeds must travel at different altitudes. The faster the orbit, the lower the altitude. So it would be impossible for the debris to "catch up" with the space stations three times. Even if the two were moving in opposite directions, they would collide once every 45 minutes, as the ISS' orbit period is 90 minutes.
Factual error: During the first few minutes, before all the comsats get taken out by the debris storm, conversation between Houston and the Space Shuttle include Quindar tones. Quindar tones are the sharp beeps attending each voice transmission. These tones are obsolete and haven't been used for many years.
Audio problem: In several scenes, Sandra Bullock used radio equipment in efforts to call for assistance. There is a sound made by vintage, mechanically tuned radios called "heterodyning" that results in a squealing, variable frequency sound in the speaker. While heterodyning added a degree of drama to the scenes in "Gravity", contemporary digital radios operate on a different, highly stable technique for tuning, and do not produce heterodyne sounds.