Plot hole: 300 years is just too long for the children to be on their own. How did they keep their clothes relatively clean for 300 years? Since the kids are playing all day they aren't out in the fields planting and harvesting crops for food, how did they eat?
Suggested correction: For the clothes, it's highly unlikely that they wore the same sets for 300 years. They are in a town/city and on a world that is mostly empty of most human life now. They can easily just find more clothing their size from other houses and even stores. As far as food, children are very good scavengers.
The issue of clothing is not so much an issue as is the issue of food. Given that the children are growing at an incredibly slow rate, their metabolism is probably much, much, much slower and would require far less sustenance.
Continuity mistake: When Spock races back into the phaser control room, he places his electronic tablet on the console and kneels to check on the unconscious Styles and Tomlinson. From one shot to the next, the tablet changes its position on the console behind him. (00:40:00)
Character mistake: Spock tells Kirk that without being able to test their vaccine with the ship's computers, "it could be a beaker of death." The vessel that the vaccine is contained in is a flask, not a beaker. The ship's science officer should know the difference. (00:37:25)
Continuity mistake: After their life threatening surgery, Spock and Sarek somehow manage to switch beds between being in surgery and recovering. (00:45:00)
Character mistake: Kirk is surprised by Spock's behavior, but shouldn't everything about Spock be in his records? Kirk should therefore know about Spock's needs and have planned for them.
The Man Trap - S1-E2
Other mistake: When Crater is hit by the phaser, his collapse is sped up. (00:36:50)
This Side of Paradise - S1-E25
Continuity mistake: As Spock beams up from the planet, Kirk, who is manning the transporter controls suddenly pulls a steel pipe from nowhere. (00:33:55)
