Character mistake: At the start, Captain Janeway reveals that she has discovered Tuvok's birthday. Then she says it's not long before he hits the big 3-digits (meaning 100 years old). However, Tuvok was 107 when he infiltrated the Maquis and they've already spent 6 years on Voyager. Even if she didn't discover his birth year and was only guessing. Tuvok would not have replied with "indeed" but pointed out she is wrong.
Character mistake: When the now human Borg children are sculpting, one of the children tell Seven he made a cube that's precisely 1/1,000 the size of a Borg vessel. Borg Cubes are described as 3 kilometers in length, if not bigger. 1/1,000 of that would be more than 9.8 feet in length. 1/10,000 the size would have been a more accurate description.
Character mistake: B'Elanna replicates and assembles a television from 1956, including a remote control. Tom however says they didn't have remote controls in the 50's. The first wireless remote was invented in 1955 and the "Zenith Space Command" (a remote looking similar to the one in use) came out in 1956, which fits the timeline. Tom is vastly aware of 1950's technology (he seems to know when the television remote came out) and wouldn't make that mistake or B'Elanna would have corrected him given how much research she had to do to set the whole thing up.
Plot hole: When orbiting the planet, Seven says that for every second on Voyager, nearly a day passes on the planet and later Janeway says 3 seconds is nearly 2 days. However, throughout the episode, time seems to move much faster on the planet to fit the plot. It would take more than 6 days for 1,000 years to pass. And in a few hours (3) less than 20 years would pass (hardly enough time for the rise and fall of a civilization). For example, when they receive the transmission, they slow it down and immediately start playing it. The next scene the senior officers are listening to it and the Doctor says nearly a century has passed, but there's no reason (or indication) that they waited almost 15 hours to listen to it.
Plot hole: Gunfire damages the holodeck controls, and the computer announces that safety protocols are now offline. This implies that they were online before the gunfire. If they were still online before the gunfire, the bullets would not have damaged the controls.
Other mistake: As Captain Janeway enters engineering after her sonic shower incident you hear crew members calling out problems on the ship. One mentions something about Deck C and another mentions a problem on Deck 22. Letters are not used to designate decks and there are only 15 decks on Voyager.
xx:xx:xxChef Greg Swagler
Factual error: When the alien contacts Voyager after his catapult jump, he states that he is 5,000 light years away. Despite the distance, he and Janeway have a real time conversation. With this incredible communications technology, Voyager (which is at this point less than 55,000 light years from the Alpha Quadrant) should be able to have several back and forth conversations a day with Starfleet.