I wanted to base it on the Doctor from Doctor Who but the only Doctor I don't yet have is 12 and she's not quite right for him (her extroversion is waayy too high for that grumpy old Scotsman who doesn't understand social interaction)

And then I thought well hang on those stats suit the TARDIS quite well. So yeah she's based on the TARDIS, a sentient, telepathic bigger on the inside time and space machine shaped like a 1960s police box.

Time
And
Relative
Dimension
In
Space

"It means life."- the Twelfth Doctor

The TARDIS has been there from the beginning and I think that's beautiful

The Eleventh Doctor: You didn't always take me where I wanted to go.
The TARDIS: No, but I always took you where you needed to go.

Security guy: Are you responsible for this box, sir?
Fifth Doctor: I try to be.

"Something old, something new, something borrowed and something blue."
(Amy, about the TARDIS showing up at her wedding)

"It's bigger on the inside!"- multiple people
("Is it? I hadn't noticed!")- Tenth Doctor

"It's smaller on the outside!"- Clara Oswin Oswald

Bill: It's like a-
Twelfth Doctor: Spaceship?
Bill: Kitchen!
Doctor: What?
Bill: Like a really posh kitchen, all metal.

Companions being shocked by the TARDIS' dimensions is like a key part of their introduction. There's the classic "takes a step inside, backs out, runs all the way around to check they're seeing it right then goes back in while the Doctor watches in amusement". And then there's Bill just not realising for several minutes because she's only seen the TARDIS exterior against a wall so she just assumes she's walked into a really weird room.

Also the Doctor calls her "old girl" and I love it

Bill: So the TARDIS has dresses and likes a bit of trouble? Yeah, I think I'm low-key in love with her.
Twelfth Doctor: Me too.

Rose: What's a "police public call box"?
Ninth Doctor: It's a telephone box from the 1950s. It's a disguise!
(This was in 2005 but he sounded so proud. I hate to break it to you Doctor but a police box is not a disguise in the 21st century.)

The TARDIS being shaped like a police box is a really cool thing. The reason behind it is essentially that it was meant to change its appearance to blend in with its surroundings using something called the "chameleon circuit" (why an alien species would name their circuit after an animal from Earth is beyond me but hey maybe that's just the closest possible English translation. Or maybe the writers didn't think that deeply about it) but after landing in 1963 and disguising itself as a police box, the circuit broke and it stayed in that form. There have been a couple of half-hearted attempts to fix the chameleon circuit but by this point both the Doctor and the TARDIS have come to love the shape and don't want it to change. Of course behind the scenes the initial reason was probably budget. And then of course it became iconic.

Gabby: The TARDIS?
Fifth Doctor: My ship. You'll love it, Gabby, its one of the best pieces of nonsense I have.
(From the Big Finish audio drama 'Classic Doctors, New Monsters: Fallen Angels')

Rooms within the TARDIS include, but are not limited to:
- a swimming pool
- a really big walk in wardrobe
- a library
- bedrooms
- squash courts (at one point as many as 7)
- multiple control rooms, including ones that don't exist yet but have already been archived
- the Eye of Harmony (aka a literal star on the brink of collapsing into a black hole)
- a room full of pillars and vines. I don't know why it was full of pillars and vines. It just was.
- a lot of corridors
- seriously so many corridors