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: It is claimed that a meteor shower has been diverted towards the Wheel by a star in M13 (the Hercules Cluster) going nova. As M13 is a globular cluster in the galactic halo, some 34,000 light years away from Earth, it would have to be one almighty nova to affect the course of a meteor shower in our solar system - due to the countless gravitational forces between M13 and the plane of the Milky Way - and been caused at least 34,000 years before the time of the story.
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.
Plot hole: How did UNIT soldiers manage to get the TARDIS through an ordinary door into the Doctor's dorm in the research centre? The Doctor's labs in the various UNIT HQs had either double doors or a removable partition wall. The research centre is supposed to be an ordinary country house with single doors and solid walls. The TARDIS is larger than the average wardrobe and, unlike most modern wardrobes, doesn't come apart so it can be got through doorways. The TARDIS is known to be a "Type 40 capsule"...but there's never been any mention of the TARDIS being an IKEA-style "flat pack self-assembly" time machine...