Goofs

Often (but specifically in the first episode and within the first few minutes) the navigation console is shown with Lt. Sulu in the left seat with an orange alien (Arex) in the chair to his right, but several times when the shot changes, an unnamed red-shirted crewman appears in place of the orange alien.

At about 4:00, McCoy appears on the bridge next to Kirk. He wasn't on the bridge before and wasn't shown entering from the turbo-lift.

When Kirk says "Put us alongside, Mr. Sulu," McCoy stands at his left, but in the next shot with Sulu in the foreground, McCoy is gone.

At 4:48, Spock reports that the alien ship's temperature is absolute zero, a condition where no molecular motion occurs and every element is solid. At 9:47, however, Spock reports in the inner chamber that air and gravity are within two points of Earth normal. The alien ship was ruptured and open to space. There would be no air remaining, and any air that might have been held inside the opened chamber would be solid. If not solid, it should have burst out into space as the doors opened.

The ship's log that Kirk records says the Enterprise is headed out beyond the farthest star of the Galaxy and that their mission is star charting.

If they're headed out beyond the farthest star, there would be no stars to chart.

The lower half of Scotty's body from the waist down isn't drawn and painted in while he's on the bridge.

After beaming onto the alien derelict, the force field surrounding Kirk is partway off him as if the animator didn't center Kirk into the outline.

At 10:52, the malevolent entity attacks the doors of the inner chamber with energy. The lines of force do not match the lines of the door.

When the alien doors close on the landing party, the life support force fields surrounding them vanish. They did not shut them off.

Spock estimates that the abandoned alien ship has been in orbit of a dead star for 300,000,000 years. Kirk exclaims that that is longer than life had existed on Earth. Kirk may have been referring to recognizably human life however, not meaning to discount the fact life has been on Earth for 3.7 billion years, about ten times longer than the age of the alien ship.