Factual error: The "whaling boat" is too small to function as such. It isn't large enough to hold a fin, let alone disassemble a humpback whale.
Factual error: When Chekov and Uhura first see USS Enterprise and then later beam aboard, the carrier shown is not USS Enterprise (a nuclear aircraft carrier), but rather USS Ranger (a conventional or oil-fired carrier). The most obvious clue is the "island", the huge structure that sits on the deck of the carrier. USS Enterprise has the most distinctive and unique island of all the USN carriers, being totally cube-like in appearance. The carrier in Star Trek IV has a more conventional style island. [It is true that the USS Ranger was used in lieu of the USS Enterprise for the movie. This is because the Enterprise was on deployment at the time. Though by the time the movie was made the Enterprise had had its distinctive pagoda-like island rebuilt by Puget Sound Naval Shipyard in 1980, the aircraft carrier served until 2012 with a still-distinctive cube shaped island.] (00:53:30)
Factual error: At two points in the film, the Klingon vessel Bounty traverses the distance from the Earth to the Sun at maximum speed, in excess of Warp 9. The latter sequence requires about 111 seconds from the time Sulu says "Aye sir, warp speed" until they reach the Sun. While there is no real science behind Star Trek's "warp technology, " the Starfleet Technical Manual provides a formula for calculating warp speed, whereby Warp 9 translates to about 136 MILLION MILES PER SECOND. The Earth is only 93 million miles from the Sun. In the apparent time that it takes the Bounty to reach the Sun, a starship traveling at such ferocious velocity would already be outside of our known solar system and deep into interstellar space.
Factual error: They are about to perform major brain surgery on Chekov, but he is not nearly prepped for such a procedure. His head is not shaved, the area is not marked or masked and it has not been swabbed with Betadine, even though they are about to cut into his skull with a bonesaw. The "they haven't had time to prepare yet" argument doesn't hold water, because the surgeon is already approaching Chekov's head with the saw. (01:24:30)
Factual error: As they descend into the park, Uhura gives a bearing and range of 283 degrees, 15.2 kilometers to the whales, which are supposedly in a marine park in Sausalito. 283 degrees would put you in the Pacific Ocean, because Sausalito bears around 360 degrees from the park. Uhura gives a true bearing to the whales, since a relative bearing would be useless once the crew leaves the park.