Lt. Romaine has a cold sore on her lower lip that comes and goes between scenes.
When Spock is monitoring and increasing pressure in the pressure chamber, at one point the level is shown as "20 atmospheres." The next view of the gauge shows 30 atm, but the shot after that is back to 25 atm when it is supposed to be a constant increase.
When Kirk commands "Hard Port, 117 Mark 2", the Enterprise turns left (i.e. port) looking forward. The Phenomenon reappears on the rear view camera from the "right", which is port side (left looking forward; right look rearward) when it should appear in view from the "left" which is starboard side (right looking forward). When banking hard port, the Enterprise moves towards the left of the Phenomenon, so it should pursue coming from the starboard (right side looking forward) of the Enterprise which is left side when looking aft.
When Sulu observes the Zetarian community moving at Warp 2.6, Kirk replies that "no natural phenomena can move faster than the speed of light." But in the 2nd season episode, "Obsession," the predatory creature is capable of traveling at such high warp speed that it eventually outruns Enterprise.
The briefing room scene and all the scenes of Mr. Scott while Lt. Romaine is being treated in the chamber in the medical lab show him with a Sciences style chest insignia (with a stylized "globe," as Spock and Dr. McCoy would wear) as opposed to the usual "spiral" on his red duty uniform.
Kirk continually refers to Lt. Mira Romaine as "girl". But both Kirk and Romaine are played by actors in their 30s. If she were perhaps in her early 20s it would make sense for Kirk to refer to her thus. But being nearly the same age, a man calling a woman "girl" is patronizing and not a characteristic of an honored Starfleet officer. This may seem patronizing to early 21st century viewers, but it was commonplace in the 1960s when the show was made. And who can say what the cultural mores around such matters will be in the 2260s?
Incorrectly regarded as goof:
Agreed this is Lt. Mira Romaine's first space voyage. But she had not been assigned to work on the Enterprise. She is a specialist being transported to Memory Alpha to perform important duty adequate to her rank in star fleet. Also is she in her 30s and must have had a career before being promoted to the rank of lieutenant.
The entire time that Lt Romain is in the chamber floating in zero gravity, her hair is hanging downward due to gravity.
After the first lights attack, a close-up shot of Sulu is reversed: his hair is parted the wrong way.
When Lt. Romaine is "floating" in zero-g while in the hyperbaric chamber, the platform supporting her is visible beneath the small of her back.
At around 37 minutes, the main cast walk past a crewman who appears to be turning an invisible wheel. However, since HD versions of this episode became available, it is possible to see that he is turning a partially transparent tool, presumably made of acrylic.
At 48:00 (for 10 seconds) Kirk, McCoy and Spock are speaking. Through the doorway behind McCoy, you can see a discarded newspaper lying on the floor that one of the (TV) crew members apparently left on the set. It is still there at 48:58 for another 25 seconds or so.
Memory Alpha is a vast database of knowledge accessed freely throughout the universe. It is inconceivable that such a huge amount of data had no backups made. Even in the 1960s, when Start Trek was filmed, it was standard practice to have at least one copy of all data.