Plot hole: The Doctor and Romana comment on a recording which has finished playing before they have entered the room. Not only that, but all the other characters behave as if the Doctor and Romana had seen this recording.
Revealing mistake: In episode 1, watch out for the wires pulling K-9 along the beach. (Unlike a real dog, K-9 didn't take well to the soft sandy surface and sank into the sand. Pulling K-9 along on wires provided an illusion of movement when the motor inside wasn't up to the job).
Continuity mistake: In "The Deadly Assassin" the Doctor's blood is red but in "State of Decay" his blood is suddenly dark blue. Then it's back to being red in "The Two Doctors."
Revealing mistake: The Grove is one of the most obvious indoor sets masquerading as a lush outdoor environment. The sound the actors' feet make on the floor is especially unforgivable.
Revealing mistake: When the Keeper is on the TARDIS, he asks to borrow the screen. But on the upper left corner, you can see it over where it's meant to be.
Continuity mistake: The Doctor somehow manages to secure his scarf across the catwalk, despite having only gone on seconds before the Master. He then manages to put it on again during the seconds it takes for the Master to alter the angle.
Answer: TARDISes are generally available for properly authorised use on Gallifrey; they're not usually assigned to a particular Timelord on a long-term basis. The Doctor stole his when he left his homeworld.
Tailkinker ★