Factual error: Gary Sinise looks at a computer screen with a short section (about two full twists) of DNA on it and proclaims that "This DNA looks human." He could have been looking at DNA from any single-celled organism and it would have looked just as human as what he was looking at. All mammals have 90+ percent of their DNA in common, he would have to have sequenced the entire DNA strand (something like 3 billion pairs of amino acids) to identify it as human, something that would be totally beyond the capacity of anything but a well-equipped genetics lab, something they show no sign of having.
Factual error: Their complete and utter disregard for the most basic scientific facts in this movie is amazing. It's already been said that Gary Sinise couldn't possibly have recognised the DNA sequence as human (that segment may have been enough to produce a single protein common to any lifeform). The thing that gets me is that he recognises that the DNA is missing a couple of "chromosomes" to complete it. DNA is made of units called nucleotides (remember A,T,C,& G?); chromosomes are formed by huge strings of DNA wound together (not the other way around). You don't need a degree in Biology to know this, you just need to have stayed awake in high school.
Factual error: The atmosphere pressure of Mars is 0.07 Bar, Earth is 1 Bar, about 14 times greater. Yet the plastic covering of the greenhouse is shown flapping with the outside breeze. With the inside of the greenhouse having an Earth-like environment its plastic covering would have been inflated like a balloon against the weaker Mars atmosphere.
Continuity mistake: When the three astronauts move from the "white room" into the viewing room, they all take their helmets off and carry them into the viewing room. They watch the whole 3D show while holding their helmets, but when the show ends, they are no longer holding them, and when the countdown starts, all three helmets can all be seen on the floor back in the white room. (01:26:40)
Factual error: The crew of the rescue mission abandon ship after the engines explode. They then rendezvous with a supply module that's already been in Martian orbit. The problem is that the engines exploded just as they were attempting an orbital transfer a from solar orbit to a Martian orbit. As they were unable to make the necessary burns to slow down and correct their angle to complete the transfer, they wouldn't stay in the vicinity of Mars for very long at all.
Plot hole: When Tim Robbins is floating away in space his wife could have easily saved him. She could go out a little further, use the grappler to catch her husband, then use the fuel she has remaining to halt their acceleration towards the planet and return them partway to the necessary orbital altitude/velocity to rendezvous with the resupply module. Then, Gary Sinise could come out partway to them using his thrusters, Connie could shoot the grappler out to him, and he could reel both Connie and Tim Robbins back in to the resupply module.
Factual error: The supply vehicle landed by a parachute as it is seen deployed on the ground. Mars has an atmosphere that is approximately 1% as dense as that of the Earth. The parachute shown is nowhere near large enough to slow down the landing. In fact, it is greatly smaller than a chute needed to land even in earth's atmosphere.