Plot hole: Trelane says he studied Earth images that travelled to him at light speed, and earlier they establish Gothos is 900 years from Earth. But Trelane references Napoleon and Hamilton, who weren't around until 1800 or so. That would put this episode in 2700, but the original Trek episodes are set in the 2200's.
Continuity mistake: When Kirk and Sulu enter the records room, they pick the lock. Later when they beam the officer back down, he enters the room without unlocking the door. The room should be locked since they beamed him down in the "past" erasing their having been on Earth and in the records room.
Factual error: Towards the end of the show the Enterprise is leaving Earth orbit and heading towards the sun. We see the Earth diminish and the moon appear looking exactly as it does from Earth. From this angle we should be seeing the "dark side" of the moon, which looks completely different.
Other mistake: Spock and McCoy have a brief argument over whether Charlie could have survived on his own all those years. Because they're talking at the same time, Spock saying "Doctor" as McCoy says "Mr. Spock," the closed captioner apparently misunderstood McCoy's line. The caption has him calling Spock "Dr. Spock."
Factual error: When they are searching for Finney hidden somewhere in the ship, all ship noises are deadened, and the heartbeats of those on the bridge are muffled by McCoy. In order to locate Finney's heartbeat the ship's auditory sensors are magnified by "one to the fourth power". That's 1x1x1x1 = 1, ie no magnification.
Factual error: Edith Keeler says to McCoy, "My young man is taking me to see a Clark Gable movie." In 1930, Clark Gable was an uncredited bit player who had last made a film in 1926 - Edith, a New York charity worker, would not even have encountered his name.