Factual error: Mr. Spock states that the asteroid is almost as large as the Earth's Moon. Such big bodies must be predominantly spherical due to their own gravity. However, the asteroid is very elongated, with a very irregularly shaped surface. (00:06:40)
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)
Continuity mistake: When Miramanee and her companion first see Kirk at the obelisk, the two women are standing, holding their fruit baskets, just inches from each other. In the very next, reverse angle shot, they're suddenly several feet apart. (00:09:10)
Continuity mistake: When Spock orders the phasers to fire, they shoot outward from the ship at an angle from each other, but when they hit the asteroid they converge at a single point. (00:24:00)
Continuity mistake: When the Enterprise is trying to push away the asteroid using the deflector, a quick shot of Uhura is shown from stock footage of "And the Children Shall Lead," but she is not in the rest of the episode.
Plot hole: If diverting the asteroid was so important, what was the crew doing on the planet to begin with?
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.