Is it theoretically possible to have a larger inside volume than the outside exterior? Last night I was listening to the audiobook of Harry Potter and the Chamber of Secrets (yeah, I know..) and it talked about the trunk of their enhanced car that was much bigger from the inside, than it looked from the outside. It made me wonder..
I'm not a scientist, but have a very basic understanding of physics. Continuing assuming magic isn't real and all theories that yet haven't been proven false, such as parallel universes and string theory (just random chosen subjects), may apply, would it theoretically be possible to have a larger inside volume than the physical object itself. 
Basically, with some fantasy: Is it theoretically possible to fit an elephant in a mouse-sized box?
 A: The question is a little hard to define because, as pointed out in the comments, it's not clear what the "outside volume" is, or how to define it in curved space to compare it to regular space. Still, I think there is a sense in which the answer is "yes".
A regular sphere has area $A = 4 \pi R^2$ and volume $V = 4 \pi R^3/3$, giving a relation $V = (1/6)\sqrt{A^3/\pi}$. We may ask whether there is some situation where a sphere would have a larger volume for a given area. If we imagine that someone standing on the surface of this sphere can only measure the area, they would infer the volume using the above formula, and they would be surprised to know that the interior volume is in fact larger.
There is a situation in which this can happen. For many years, cosmologists thought that the most likely shape for our universe was that of a 3-sphere, which at a fixed cosmological time has a metric given by
$$ds^2 = \frac{dr^2}{1-r^2} + r^2 (d\theta^2 + \sin^2 \theta d\varphi^2)$$
in suitable coordinates. The area of a sphere at a given radius is still $A = 4\pi R^2$, but following the standard methods of differential geometry the volume is
$$V = 4\pi \int_0^R dr\ \frac{r^2}{\sqrt{1-r^2}} = 2\pi \left(\arcsin R - R \sqrt{1-R^2}\right).$$
You can see by plotting that for any $R$, this volume is larger than the one given by the usual formula, even though the area is the same.
Edit in response to your edit: you ask whether it's possible to fit an elephant in a mouse sized box. Unless our conception of space radically changes (again), then the answer is clearly no. A mouse sized box is a box in which a mouse fits, i.e., its interior volume is that of a mouse. You're asking whether an object can have an interior volume larger than its interior volume; I hope it's clear that this is not possible. You can, however, fit an elephant in a box with the surface area of a mouse.
A: Surprisingly, the answer is yes. Or at least, yes, if you rephrase the question a little bit.
As you might know, gravity can deform spacetime. (actually this is not completely true, it is actually the deformation of spacetime, which we call gravity) This means that around heavy objects, distances or timescales, can become longer or shorter. This will form the key to the answer to your question.
But let's first take one step back. The question is whether the inside volume can be larger than the outside exterior. One can clearly define the size of the inside volume just by taking a block of 1 m³ and measuring how many of these blocks fit inside the volume. However, the size of the outside exterior is not so clearly defined. The best one can do is to measure the lengths and surface areas and the shape of the outside of the volume and then use your knowledge about other volumes with the same shape to infer the volume. For instance for a cube, we know that the volume is typically given by  a³, when a is the length of the sides. However, this is only true for flat space.
If one puts a mass inside the cube, the mass will curve space in such a way that this relation will no longer hold. This means that somebody on the outside, measuring the sides of the cube and from this calculating the volume using the stated relationships, would arrive at a different volume than a person stacking 1 m³ cubes.  
In 3D this is very difficult to imagine, but in 2D it is easy. Just replace the cube in space by a square on a surface and compare the results for somebody measuring 1m² squares and somebody doing the calculation from the length of the sides. If you do this on a curved surface, like the surface of a sphere, the two measurement no longer agree. In reality mass causes this curvature.
A: The physical concept most closely mimicking the experience of fantasy items like bag of holding is a traversable wormhole. Essentially, it is a connection between two different regions of space, a portal (yes, like in a videogame), or maybe even bridge between different universes.
In the former case the inside volume of a car's trunk would be located elsewhere and simply accessed through the trunk's door which would be the actual wormhole throat.
In the latter case the 'inside volume' would be (potentially infinite) the other universe or it could be a small closed universe like in some sci-fi settings, for example, see the picture from 'Orion's Arm' site: 
So the question is, do such traversable wormholes exist? Mathematically, it is quite simple to write a metric for such a space: one simply 'glues together' two sufficiently separated regions of space along two copies of some surface, which becomes a wormhole throat and then smooths out the resulting singularities. One could even avoid things like tidal forces, regions with dilated time etc. or localize them so they won't interfere with two-way travel through such a wormhole. But from a physics' perspective such construction may very well be impossible, and here's why. 
Einstein's equations (which are the main equations of General relativity) have the following structure:
$$
G_{\mu\nu}= \kappa T_{\mu\nu},
$$
where the left hand side is purely geometric, while the right hand side is dependent on matter. So if one would take a space-time geometry with truly amazing properties like time-machine, superluminal space travels or traversable wormholes, the right hand side would be corresponding to so-called exotic matter, and hence not compatible with the ordinary matter from which our every-day world is composed.
Most notably, one of the requirements for such exotic matter seems to be always some sort of  negative mass. And while Casimir effect may offer some hope, for now at least it is more or less pure speculation.
A: I'd like to add another option to the answers provided by Javier and Crimson. There is a spacetime geometry popularised by Wheeler (I'm not sure it originated with him) called the bag of gold spacetime. This doesn't have a Wikipedia page I can link to but I have mentioned it in my answer to Creation of a miniature universe? There are a few other related questions on this site as well.
The spacetime is constructed by patching a regular Schwarzschild geometry to a geometry looking a bit like an expanding universe. Mathematically this is a perfectly reasonable thing to do, but whether the mathematics have any physical relevance is highly debatable. However the idea has some history. The Russian physicist Markov pursued this idea quite vigorously and I have provided links to papers by him here. More recently the idea was used by Lee Smolin as the basis for his ideas on black hole evolution. I should probably note that both ideas are not accepted by the mainstream community.
Anyhow a geometry like this would appear as a sphere to the observer outside it, but anyone passing through the sphere would find a new region of spacetime that could be arbitrarily big. So there would be plenty of room for an elephant in there.
The only problem is that (I think, though I wouldn't swear to it) the entrance acts as an event horizon similar to the one around a black hole. That would mean you could drop things into the bag but you'd never be able to get them out again.
