Audio problem: If you listen carefully to the soundtrack of part four of "The Moonbase", some 'chatter' and 'feedback' from the floor manager's headphones can be heard. (At the time this particular episode was recorded, there was just a one-week gap between recording Doctor Who and transmission, so there was no time available to re-record the episode, and this mistake had to be left in).
Factual error: During every outdoor scene in Antarctica, heavy blizzards are shown raging constantly. In reality, such blizzards are extremely rare at the South Pole, thanks to the almost total absence of precipitation Antarctica is effectively a desert, as the air is very dry: with very little moisture in the air, rain or snow is almost impossible.
Continuity mistake: The rhyme the Doctor recites in episode 3 is different from the original rhyme heard in episode 1. (Apparently, the Doctor [William Hartnell] forgot his lines in episode 3, when he came to the point of reciting the rhyme. So he just improvised what he thought they were, rather than what had been scripted).
Revealing mistake: The Daleks that appear in this and other Doctor Who stories were rather expensive, so some ingenuity was required to make the small number available look like they were more than they were. Tricks employed in "The Power of the Daleks" included making four Daleks look like an army by having them circle around the set a few times (note the delay between the fourth Dalek exiting and the "fifth" Dalek entering), and the blatant use of photographic blow-ups in place of "real" Daleks.
Factual error: In the first close-up view of Mondas, the planet is spinning rapidly so that we can see that its continents are identical to Earth's. But this would mean that Mondas is spinning on its axis once every few seconds - tens of thousand of times faster than Earth. This would be such a fast rotation that the planet would not be able to hold together under the centrifugal force.