Factual error: The last time Eleanor Arroway talks to S.R. Haddon, he's aboard the Russian space station Mir, and Haddon explains that he's up there because the "low oxygen" and zero gravity counteracts his cancer. In fact, there is no "low oxygen" environment aboard space stations or other spacecraft. Low oxygen content would, of course, kill any astronauts or cosmonauts in short order. The breathable air in spacecraft always has at least the same oxygen content as Earth atmosphere at sea-level. In fact, most Russian missions used excessive amounts of oxygen. S.R. Haddon's original dialogue was probably "high oxygen and low gravity," but the line was bungled and allowed to remain in the film.
Plot hole: After Ellie makes her journey which she firmly believes happened she is questioned as everyone else watched the pod drop straight down into the water. But no mention is made of the chair that is violently shaken loose during her 18 hour journey. If that were to have happened in the split-second the pod fell through she would have been badly injured more than just a bump on the head.
Factual error: The scene where Ellie and Palmer are looking at the stars next to the Arecibo observatory dish, they are looking at the constellation Cassiopeia. This constellation is visible to the North. However, judging from the parts of the antenna structure visible in the background, they are actually looking South. The suspended walkway/cable raceway connecting the control room and the antenna structure visible in this shot is on the North side of the dish. In real life, the control room and observation areas are on the North side of the dish. It is also not clear what time of the year this was but Cassiopeia is not visible from Puerto Rico during part of the year due to its low elevation above the horizon.
Factual error: Hokkaido island geography shows a relatively smooth coastline. The location of the second machine cannot be natural, but must have been added post production.
Continuity mistake: Ellie tells Palmer her father died when she was nine years old. Later, when Mr. Hadden goes over her life history to impress her, the dates of her birth and her father's death indicate she was ten years old when he died.
Continuity mistake: While receiving the prime number pulses, there is a discontinuity between the pulse sequence being received and what is displayed on the monitors. As the sequence for "seven" starts (right after Ellie tells Fish to start counting the pulses), the camera switches to a computer monitor display which indicates the sequence for "seven" has already completed, and is logging the pulses of the sequence for "eleven".
Continuity mistake: When talking with Ellie on his plane Haddon's hand alternates to being on Ellie's shoulder in shots facing him to not touching her in shots behind him.
Character mistake: When they first receive the alien signal, they decode it and play it as a TV signal which shows Hitler speaking at the opening of the 1936 Summer Olympics in Berlin. Kent translates what Hitler is saying as "I declare the Games in Berlin, at the celebration of the first Olympics of the New Era, as opened." Hitler actually says "the *eleventh* Olympics of the New Era." The "New Era" doesn't refer to some Nazi concept, it's used to separate the modern Olympics from the ancient ones in Greece.
Suggested correction: Also, being terminally ill, he is grasping at straws, and can't be expected to think solidly. After all, gravity wouldn't mean all that much to cancer cells in the human body, and special oxygen conditions of any kind can be generated on Earth (there is a weak possibility that low but sufficient for a human oxygen levels would slow cancer cells, which are often less efficient, more vulnerable in certain areas than healthy cells), so no need to go up, and finally, he may simply be Dennis Tito-ing and making an excuse to do so.
dizzyd