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)
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)
Continuity mistake: When McCoy places the old tricycle down on the ground, the handlebars are facing the right with its wheels facing the left. When McCoy is attacked by the diseased Only, Kirk and Spock subdue him, and as he lies on the ground the tricycle flips over between shots, so the handlebars suddenly end up facing the left. (00:04:30)
Factual error: If Miri's planet is a "duplicate" (meaning identical) Earth, it should have clouds. It doesn't. This remains a mistake because "duplicate" means "exactly the same," and thus the clouds should be there. The special effects crew forgot to put them in. Noteworthy: the very first thing fixed in the digitally enhanced version of this episode was the duplicate Earth. It has clouds now. (00:01:30)
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.
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?