Do we have any idea how big the interior of the TARDIS is?
Is it even a fixed size, since it was recently renewed?
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Short version: No, all we know is that it's bigger on the inside.
Long version: The exact size, shape, and layout of the TARDIS has never been fully described, and the layout can change depending on season, and the writer. The only fixed room in the TARDIS is the main control room (the room with the central console that is seen in almost every episode). Even that room changes, usually getting a new layout, or slight improvements when there's a new Doctor (occasionally changing mid-Doctor, though).
There are a few other "known" rooms, including a Library, Wardrobe, Cloister Room/Bell (the Cloister Bell sounds when danger is imminent), a Holding Ring (which is where rooms of past companions could be found), and a Swimming Pool. (Source). Additionally, there's various storage rooms, and it's implied that each companion, upon joining the Doctor in his travels, gets a room to store all their stuff. There's also rooms that store food, a few bathrooms, lots of corridors, medical bays, and feasibly any number of other rooms used for any number of other purposes.
Additionally, there is at least one secondary control room used for a period of time by the Fourth Doctor. In The Doctor's Wife, the TARDIS mentions that it has "archived thirty control rooms", only 12 of them had happened yet. More details on the different control rooms can be found here: TARDIS Control Rooms
The only hard fact we have about the interior of the TARDIS is "Bigger than the outside."
It's been said several times that the Doctor can control the configuration of the TARDIS interior, and he has in fact changed it several times ("You've changed the desktop theme" from the 10th Doctor's Children in Need special).
The inside of the TARDIS has no real effective limit - we've seen tiny containers with space for millions of Daleks in the new series, and we know that (this being Doctor Who) anything that's stated is subject to future retcon.
I'm not sure how accurate or serious was the Doctor, but in Journey to the centre of the TARDIS, the Doctor states it is infinite:
-Picture the biggest ship you've ever seen. Are you picturing it?
-Good. Now forget it. 'Cause this ship is infinite.
Edit: But the Doctor always lies.......
Edit 2: Buuuut I don't see a reason why the Doctor would want to lie about that.
In a discussion at the Permuted Press forums, Peter Clines said:
"The War TARDIS idea gets a lot of use in the books that revolved around the Time War (well... around the original Time War), like The Ancestor Cell. One of the funny bits is that War TARDISes appear as massive, planet-sized battleships. The Doctor's companions are terrified, but he brushes it off by explaining the Time Lords are just showing off and using the chameleon circuits to make the outside of each TARDIS match the inside."
I've yet to read that novel, but as it's an Eighth Doctor adventure, its canonicity is in question.
In the episode Journey to the Centre of the TARDIS, don't forget that the Eye of Harmony, or the main power source for the TARDIS is literally a decaying star sitting in what I can only describe as a Dyson sphere.
Big? Big isn't the word for it. Fracture the surface of the Earth? I'd say closer to crushing it and terminating every speck of life on it.
Well, the Doctor once had to jettison a quarter of it (the TARDIS). It was infinitely large, which caused a bit of cognitive dissonance for Adric. ("Castrovalva", 19th season, first serial, January 1982) That the jettisoned portion contained the Zero Room caused quite a bit of physical dissonance for the Doctor. (Then, bizarrely: Carpentry to the rescue!)
According to the 1983 Doctor Who Technical Manual, the TARDIS interior is infinite in size. However, this would seem to be contradicted by statements in the show, including ones from the same era. The most notable problem would be with the decision in jettison one quarter of the TARDIS in "Castrovalva," which does not really make sense if the ship is actually infinite.
I asked Peter Davison about this at a Doctor Who convention, and he said that, as far as he was concerned, the TARDIS was very large, but finite in size. "Big, big," was the way I think he phrased it. He blew off whatever the Technical Manual said as irrelevant cruft. Although I doubt Davison had ever looked at the book, dismissing it was probably not a terrible idea, since it contained some other obvious howlers—like placing the kaled mutant at the very bottom of the dalek interior it showed.
The question of size and mass makes sense in universe. Time Lords can alter the size (dimension) of the TARDIS irrespective of the mass of the TARDIS. (Time Lords being dimensional engineers after all!) So while the TARDIS has infinite volume it does not necessarily have infinite mass. This then allows for them to eject a quarter of the mass of the TARDIS even though it has an infinite volume in it's interior.