Revealing mistake: When Kirk and Miri are in the kid's hideout, Kirk is attacked by Louise, the wild woman. After Kirk knocks her off his back, she falls to the ground and tries to get up again, Kirk has to resort to stunning her with his phaser. The blue beam starts coming out then starts coming back towards the phaser before going back out again. (00:26:12)
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: The ropes tying Yeoman Rand to the chair keep changing positions (above her elbows or below her elbows) between shots. (00:39:40)
Factual error: In the opening scene on the bridge, when Spock states the planet's properties, the circumference is given in US miles; the mass is given in metric tons; the density is given in metric grams per cubic centimeter; and the atmosphere is given as oxygen/nitrogen. No scientist of Spock’s standing would mix US and metric unit systems. The atmosphere composition should also be stated reversed as “nitrogen/oxygen” with the most abundant gas first. (00:42:00 - 00:59:00)
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.
Factual error: If this planet is a duplicate, an exact copy, of Earth, then Lake Okeechobee, the largest lake in Florida, is wrong. Seen from space, it is much larger and more distinct than the lake shown on this duplicate.
Other mistake: In the opening, the planet looks to be turning the same as Earth does. On the viewer it is turning correctly. However, it is too quickly to be accurate, assuming it is a copy of Earth. If so, at this higher speed, the days would be shorter. Also, the ship should be moving at such a speed as to see the planet moving in the opposite direction.
Visible crew/equipment: When Dr. Van Gelder tells Kirk that he wants asylum and his word that he won't be taken back to Tantalus, in the wideshot from behind Kirk as he moves to his right the two tape marks can be seen on the floor beside Van Gelder's feet.
Plot hole: For a penal colony, Tantalus V has really poor security. Van Gelder manages to escape with ease, by hiding in a box - the only box - beamed up to the Enterprise.
The Corbomite Maneuver - S1-E11
Other mistake: When Kirk is in sickbay with McCoy he uses sickbay's desktop monitor to contact Spock on the bridge, then when Kirk is in his quarters he uses his own desktop monitor to contact Spock again. Both have the same paint dings and smudges around the console screens, revealing the same one was used during filming. (00:04:30 - 00:06:30)
Suggested correction: That might only true in today's standards. But we have no idea what future generations will choose to make standard.
Bishop73
This is such a trivial criticism that it should be removed to be fair. Whatever measurement standard is used in the future, it will be uniform without mixing of different unit systems.
Ken S
But that's an assumption based on what you think the future would be like. The British and Americans currently use a mix of different unit systems. While many US students use miles and pounds, they still calculate density as g/cm3.
Bishop73
No scientific or engineering facility on Earth uses imperial measures nowadays, including NASA. That is not going to change.