Factual error: When the shuttles are approaching the station, their engines are firing from the rear, meaning they are in constant acceleration toward the station. When arriving at a station, you would need to slow down as you approach, meaning the engines should be firing forward (or the shuttle should reverse direction). This happens in all scenes where shuttles are approaching the station.
Factual error: When the Armadyne computer reboots, you can see the typical bootup screen which gives away certain information about the computer: In two consecutive shots of the screen, you can see that this computer has a BIOS, a GA-MA790XT-UD4P mainboard from Gigabyte, 4 gigabytes of RAM, and apparently works with IDE drives. The successor to the BIOS is called UEFI, the mainboard and 4 GB of RAM are only remotely up to date as of 2014, and IDE drives are already dying out in favor of SATA drives. (01:39:20)
Factual error: As the shuttles are approaching Elysium Station, their engines are constantly firing. Still, bottles and other miscellaneous items are shown free-floating around the shuttles. Weightlessness exists on space ships only as long as they are coasting. As soon as they are under thrust, things "drop" to the rear of the ship in exactly the same way the bowling ball we probably all heard about in physics class rolls towards the back of the iconic train.
Continuity mistake: As we see John Carlyle code, he has a short paragraph up on his screen, and he's adding text at the end of it. In the closeup, we're treated to a very nice bit of Assembly code, and he's adding a few lines in the middle. When we see his screen from the back next, his nice short paragraph from before has grown a table on top, and he's back to adding text at the bottom. (00:40:30)