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.
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".
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: 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.