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
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.
The Doctor always lies, but I don't see a reason why the Doctor would want to lie about that.
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.
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.
It has been claimed that the interior of the Tardis is infinite. Other answers mention the doctor claiming that at least once.
Someone asked if that makes the interior of the Tardis larger than the universe.
I once read that it was possible for there to be different infinites, all infinite, but some larger than others. So Adric might have been wrong to say that a quarter of the mass of the Tardis couldn't be infinite.
At the present time it is unknown whether the universe is incredibly vast but finite, or infinite. Thus it is possible for both the Tardis interior and the universe to be infinite, and for the universe to be larger than the Tardis interior, or the tardis interior larger than the universe. If the universe is finite, it might be contained inside the Tardis interior somewhere, while at the same time containing the Tardis exterior (which is almost always within the universe) which contains the Tardis interior. And so on and so on.
And since all the other Tardises also (almost always) have their exteriors within the universe, if the universe is inside the interior of any Tardis it could also be inside the interior of each and every Tardis. So each and every Tardis could contain the universe, which contains the exteriors of each and every Tardis, and each and every Tardis interior would contain the universe and each and every Tardis exterior, and so on and so on.
I think I remember an episode, possibly "Logopolis" where the Master did something to the Doctor's Tardis, putting it inside itself. Thus when the doctor went out the door he found himself still in the Tardis, an outer Tardis you could call it, and when he went out the door of the outer Tardis he found himself in an even outer outer Tardis, and so on.
Earlier, in the 4th Doctor episode "The Invasion of Time", Sontarans pursued the Doctor into the Tardis and he led them on a long chase through brick rooms and corridors, filmed in an abandoned building. And I think that was my first hint that the Tardis interior was not merely ordinary house sized but probably at least labyrinth sized or Pentagon sized or possibly even Death Star interior sized.