Plot hole: The law officer who arrests Kirk recalls hearing Kirk call the "spirit" Bones. But he wasn't there when Kirk spoke to McCoy. He rushed into the scene several minutes later. If he's lying, how would he know that Kirk used the name Bones? He wasn't there to hear it. Even if he had been, Kirk did not refer to McCoy as Bones in that conversation.
Plot hole: In the conference room, Spock informs Kirk that his theory of a Romulan cloaking device kept the sensors from detecting the ships. However, in S1:E14, "Balance of Terror", they figured out the cloaking device, and how it worked. Unless he was feeding into the deception, Spock shouldn't have had any theory when it was a fact.
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Plot hole: When the Medusan Ambassador Kollos and Dr. Jones materialize, the latter is wearing a protective visor and Mr. Spock can see that. Spock then mistakenly addresses Dr. Jones as Ambassador Kollos. Spock's suddenly illogical mistake seems odd: why should the Ambassador need to wear a visor? Shortly after, why do both Jones and Spock wear visors while carrying the Ambassador's still closed gasket to his cabin?
Plot hole: The whole plot centers on the crowded conditions of the planet. The people are literally bumping into each other. Even if they had the blueprints to build a full size model of the Enterprise, there would be no room for it.