I know we would have no way to theorize how Timelord technology works, but is there ever any explanation given in Doctor Who about how Timelords create buildings or containers that are bigger on the inside than on the outside?
The interior of a TARDIS is actually in a different dimension than the exterior. That other dimension's contents are maintained by the TARDIS' energy. If the TARDIS is low on energy, it can delete rooms from the interior to free up energy (The Doctor's Wife).
The construction of a TARDIS isn't actually a process of building, it's a process of growing. It's unclear if this means the TARDISes are actually biological in nature, or if the technology Timelords employ when making TARDISes simply mimics biological growth (The Impossible Planet). The specifics of how the Timelords manage this feat hasn't been revealed.
From "The Robots of Death" (1977)...
LEELA: So, explain to me how this Tardis is larger on the inside than the out.
DOCTOR: Hmm? All right, I'll show you. It's because insides and outsides are not in the same dimension.
(The Doctor gets two boxes from on top of a cupboard. One box is larger than the other.)
DOCTOR: Which box is larger?
LEELA: (pointing to the larger box) That one.
(The Doctor places it on the time console then walks over to Leela with the other, smaller box.)
DOCTOR: Now which one is larger?
LEELA: (pointing to the larger box) That one!
DOCTOR: But it looks smaller.
LEELA: Well, that's because it's further away.
DOCTOR: Exactly. If you could keep that exactly that distance away and have it here, the large one would fit inside the small one.
LEELA: (beat) That's silly.
DOCTOR: That's transdimensional engineering, a key Time Lord discovery.
There actually is a way to plausibly explain, using modern human physics.
We know, through quantum theory, that the universe consists of more than just the 3 dimensions (or 4, depending on how you count Time) that we are able to directly experience. Some branches of quantum theory postulate thousands of additional dimensional axes or even higher. From our perspective, locked inside these 3 particular dimensions, those look like they're curled up at the individual points, but I'd imagine that if we looked at them outside of this context, they would just be additional axes on the grid.
Now imagine that you've found technology enabling you to shift matter from these dimensional axes to those. You could create a portal of sorts that you could walk through and be "rotated" into a separate 3 dimensional space. It's still 3 dimensions, just 3 different dimensions. Once in that separate space, you're no longer constrained by the matter in these 3 dimensions. The walls you were inside of here don't necessarily exist there. You could build a huge open space with everything you need, with this little blue box as the entrance. The TARDIS, as depicted, apparently does this so seamlessly and consistently that even light goes right through unaffected, allowing you to view directly in or out of the doorway. Time Lord technology is amazing, isn't it?
That would also explain the appearing and disappearing blue box, as well as the chameleon circuit. Since they (we assume) have the technology to "rotate" matter to and from those other dimensions, the disappearing is just the ship pulling in the gangplank, as it were. That blue box is ultimately just a solid illusion, and thus could be modified in its appearance. Once you have that first piece, the rest falls right into place.
So how do you power such a beast? Well, with so much room to play with, you could have some massive battery stores and generators. You could also have energy scoops that could be extended ("rotated") straight into the heart of a sun or other energy source to recharge the ship. Heck, you could have a star itself at the core of the ship. Plenty of power for billions of years.
But what would you use to build it? You could harvest material from those other dimensions just like you could from here (but with a bit of extra effort of course). Modern physics asserts that those are just as likely to contain matter as our own.
And as an added bonus... Since time is just another dimension, if you can move about in these extra dimensions, time travel would be a piece of cake, just like walking up the road. You just have to maintain fixed points to avoid any serious paradoxes, and Bob's your uncle. As I said, once you have that first piece, the rest is relatively simple.