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 nucleotides) 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: 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.
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.
Factual error: The young guy builds the "woman of his dreams" with M&M's in that space ship. The double-helix rotates around its own axis, though it wouldn't do so - even in zero gravity.
Factual error: There are scenes showing the astronauts battling against a fierce Martian wind. In fact the atmosphere is so pathetically thin on Mars that even a 200mph wind would feel like nothing more than a gentle breeze.
Factual error: Ship explodes. Crew evacuated to the REMO, but Tim Robbins slips from the REMO after attaching the lifeline. They tell his wife she has insufficient fuel to reach him. Velocity is determined by fuel, not distance. She could have used 25% of remaining fuel to reach him, 25% to arrest speed then 50% to get them both back to the REMO. It's not like a car - once in motion, you remain in motion.
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.
Factual error: When with the alien inside 'the face' and looking at the solar system, the solar system is rotating backwards or clockwise. Our solar system rotates counterclockwise.
Factual error: When the fuel leaks out of the fuel line, why does it freeze when it should turn to vapor? Liquid does not instantly turn to ice in a vacuum (big thermodynamics issue there), but does the exact opposite due to the lack of pressure.
Factual error: The ship is hit by micro-meteors, and a major fuel line is ruptured. Yet this loss of fuel goes completely unnoticed by the crew or by the computer. And the fuel is typically binary (fuel and oxidizer are separate), so a leak of one of the components wouldn't cause an explosion like that in space.
Factual error: When the micro-meteors penetrate the skin of the ship, the ship's computer starts failing, seemingly in time with the loss of atmosphere. Once the atmosphere loss is stopped, a simple reboot restores the computer to functionality, despite the damage.