Factual error: The Enterprise accidentally travels a short distance outside the galaxy and can't find its way back. But they'd have to travel for months to get so far outside the Milky Way that they couldn't, well, just turn around in the void and see it. Our galaxy is huge. 100,000 light years across. Very huge. And that barrier may surround the galaxy, but even it is big, pink and visible.
Factual error: When Kirk is given the medicine badge, it's a stretchy elastic/Spandex headband. This is supposedly a completely pre-industrial culture paralleling the early Native American tribes. They have no fabric, no yarn, no spinning wheels - only hides and animal hair, neither of which can stretch a la Spandex. (00:14:40)
Factual error: Spock plays a piece on a harpsichord that he says is by Brahms, but Brahms was a late romantic composer and the piece is a simple baroque dance piece. Also by the time of Brahms the harpsichord was already obsolete, a composition like this wouldn't be sitting on a harpsichord.
Factual error: In the opening sequence, Spock identifies the alien ship as possessing "ion propulsion" which he says is "unique technology." Scotty is similarly impressed and says, "They could teach us a thing or two!" Kirk later comments that "Advanced ion propulsion is beyond even our capabilities." However, even back in the 1960s, ion propulsion was physically feasible, while Warp propulsion was complete fantasy. Ion propulsion of any kind could never even reach lightspeed and would be incredibly primitive compared to Warp technology. In fact, we in the 21st Century have already developed ion propulsion, but it will probably take many hundreds or thousands of years to develop anything even close to Warp technology.