Factual error: The whole conspiracy is doomed from the start! They show a Lunar Module on the set representing Mars; the astronauts were supposed to use this to 'land' on the Martian surface. This is insane. The Lunar Excursion Module cannot operate in an atmosphere; it would burn up on entry. It has no heat shield - the engine nozzle and landing legs would have to poke through it if it had. Exterior protrusions like antennae and steering engines would burn off during descent through the Martian atmosphere. While Mars' atmosphere is thin, it is still more than sufficient to burn up a Lander designed this way. All three recent Mars Landers, Spirit, Opportunity, and Phoenix, had heat shields and parachutes to facilitate safe re-entry. This is not a radical redesign of the LEM; it simply cannot be the shape it is if it is to work in an atmosphere. Any schoolkid would know that, and the world's media would not be fooled for a second.
Factual error: If one person (at NASA) was able to calculate the distance between the returning Mars capsule and Earth based on their radio signals, then everyone with basic radio receiving equipment would be able to. The Russians routinely track American spacecraft and publish details of their trajectory in Pravda, and amateur astronomers throughout the world do so, too. This all comes from triangulation of their radio signals. There is no way that NASA could hide this simple fact from their worldwide audience.
Factual error: In the beginning of the movie, a Saturn V is seen representing the launch vehicle that will take Capricorn One to Mars. This is downright ridiculous. A Saturn V's third stage (that would insert into orbit and make the injection burn) does not have nearly enough Delta-V to make a Trans-Martian Injection from low Earth orbit.
Factual error: When Caulfield is looking at the address on the woman's magazine (when he's at Elliot's apartment) the Houston zip code is listed as 80144. Houston zip codes start with "77."
Factual error: A round-trip voyage between Mars and Earth would take about two years. Although the movie's timeline is compressed, from when the astronauts supposedly leave Earth to them supposedly returning (and dying) and then Caulfield rescuing Brubaker, the entire plot implausibly takes place in only eight months. It is mentioned at the memorial that they "left" Earth eight months earlier. The military memorial is held at least a month after the "tragedy" so that would mean they made the trip in seven months, which is impossible. The film is not set in the future when technology would be more advanced.
Suggested correction: They could use meteors that had landed on Earth. This is one of the theories for the "faked" moon landing, that they either created the moon rocks from scratch, or collected meteors. As for the death of the astronauts, that's not a plot hole, it's the plot of the movie; the powers that be wanted the men to fake everything and return as heroes. When they wouldn't play along, it was decided they needed to be eliminated.
It is clear from the narrative of the film that it was planned that the astronauts would "land" safely. Using meteorites would not work - exposure to the Earth's atmosphere would mean (and has meant) that the rocks would show weathering and chemical changes that anyone would be able to detect.