When Balok begins his 10-minute countdown of the Enterprise's destruction, McCoy enters the bridge wearing a standard long-sleeve uniform shirt. Shortly after this, camera cuts around the bridge include one focused on Spock and McCoy standing by the science station, with McCoy wearing his short-sleeve medical tunic. Then, when Lt. Bailey has his frantic outburst, McCoy goes to him to calm him, wearing the long-sleeve shirt again.
Just after destroying the space buoy, as Kirk and Spock are discussing what might be ahead of them, the two-shot of Kirk and Spock (from behind Spock) shows Spock's earpiece in his right ear. The next shot is a frontal shot of Spock, minus the earpiece.
When Kirk, McCoy and Bailey are in the transport room ready to beam over to the alien ship, they are bent over with their hands on their knees. When they arrive McCoy has his hands off his knees and slightly to the side.
When Spock puts the image of Balock on the main viewscreen, McCoy's shirt changes from a standard long-sleeved duty tunic to a short sleeve medical smock, then changes back again.
When Kirk and Spock confer, at about the 20 minute mark, Spock initially has a earpiece in his left ear. In the middle of a sentence, as the camera angle changes, the earpiece disappears.
The U.S.S. Enterprise was 289 meters in length. That is about 948 feet. That puts the length of the U.S.S. Enterprise at longer than 1/6 mile. The large alien ship was described as having a diameter of about one mile. But when the U.S.S. Enterprise is seen close up against the large alien ship, the alien ship is seen to be dozens of times longer than the former. That would put the diameter of the large alien ship closer as several miles instead of one mile.
When the Enterprise encounters the Fesarius, Spock says the mass of the Fesarius is too large to be measured. But, as a ship of exploration, Enterprise would have instruments to measure mass of various sizes of asteroids, moons, dwarf planets, planets of various types, and stars.
Spock says that the cube resembles "flypaper", and everyone agrees quickly. While flypaper will most likely be obsolete in the 23rd century, it could live on as a popular metaphor. A cursory search on IMDB reveals a number of movies titled Flypaper, which appears to cement its proverbial status in popular culture.
When Kirk is using the communication monitor in sick bay, there is a distinctive stain on the right bezel on the monitor. Later, the same stain appears on the monitor that is supposedly in his quarters.
Wires suspending stars visible when the Enterprise is being towed by the small alien ship.
The cube distance from the Enterprise is stated as 1593 meters (0.98 mile) but the visual on the outside shows it a lot closer than it should be. The Enterprise and the cube were both stationary.
When Mr. Spock points at Bailey's console light, he says, "All decks have reported green" yet the light he points to is red.
When Little Balok laughs, we see "primitive" 20th century teeth fillings, very unlikely in an advanced alien being.
Sulu hears Kyle say, "They now have one minute." Then he looks
up toward the screen and then back down toward his console. He is responding to a line of dialog from Big Balok ("You now have one minute!"), which can be heard in the preview trailer but not the episode itself, having been somehow lost in the editing room along the way.
When Uhura announces she is getting a weak signal from Balok, Spock turns from her to go to his station. When he does you can hear the floorboards supporting the set make a big squeak.
Sulu uses the ship's intercom to call Kirk to the bridge at the end of the teaser. Yet, in the next scene, Kirk is unaware that there is an emergency until he sees the red alert light flashing. No explanation is given why he doesn't hear Sulu's call before then.
In the conference room, Bailey starts to comm an order for the phaser crew to stand by, before Captain Kirk cuts him off. But no one tells the phaser crew to ignore the interrupted message and no one from the phaser crew comms back to find out what to do.
Balok makes a metaphor based on an Earth novel, "The Strange Case of Dr. Jekyll and Mr. Hyde". He had no knowledge of Earth until his belligerent encounter with the Enterprise a few hours ago, and most of his time since then has been spent studying the ship's technical manuals, which would be unlikely to contain this expression.
Even though it's painted that Bailey is panicky and over-reactive to their situation, his emotional state is the most realistic of someone encountering the unknown. It is very doubtful that the rest of the bridge crew would be as level headed and calm as portrayed. They are encountering an alien life form which says that it is going to destroy the ship in ten minutes. There would be a lot of distress on the bridge.
During emergency alert, crew members wouldn't be casually walking around the passageways while the ship is under attack. We see some of them knocked around when the Enterprise escapes from the tractor beam from the alien ship. They should be at their work places or at least somewhere secure on the ship at that time.
When the Enterprise first encounters the Fesarius, Spock states that the vessel size, "Must be a mile in diameter. Over 5,000 meters away." This mixing of measurement systems is inefficiently confusing, as the use of non-metric measurements would be totally obsolete.
Lieutenant Dave Bailey says that fear comes from the "adrenaline gland". The correct name is adrenal gland, with adrenaline being the hormone produced by it. (Note: Considering the percentage of the viewing audience who had been uneducated back then, "dumbing down" the dialog was often deemed necessary).
Just after Kirk attempts to bluff Balok with the description of Corbomite, Spock refers to his parents in the past tense, as if they are dead. In Journey to Babel (1967), Spock's parents are revealed to be very much alive, and actually make an appearance.
Upon leaving a sick-bay physical, Kirk walks down crowded corridors and into the turbo-lift in his bare socks and shirtless. He then gives the voice-command of "Bridge" as his destination. Even though he's next seen in his quarters, it would be totally inappropriate for Kirk to walk onto the bridge half-dressed.