[Editorial note: I'm going to revise this question according to the suggestions made in comments and answers during the next few days. Thank you all for your valuable hints.]
There's the "ant on a ballon's surface" story that's often told if it comes to trying to unravel the knot in people's brains due to this (alleged) contradiction. But it leaves out a whole dimension of space. That always was a trick too easy for my mind.
The final trigger to bring this question up here was a lecture called Physics in the Theatre: The mysteries of our dark universe on that matter (in German, no translation available), by Prof. Dr. Matthias Bartelmann. During the Q&A, starting at 1:24:12 he admits that even he doesn't have a better picture for "finite and unbounded at the same time" than the surface of a sphere.
Please note that the following is just for an analogy. It is not meant to be a new thesis, or even theory outside mainstream physics. (If you steal the idea and make it to the Nobel Prize with it, be assured I sue you. BTW, have you read the disclaimer? ;)
My assumptions are:
There is a question here at physics that mentions that Einstein, in his book about relativity:
[...] says that his theory predicts the shape of the universe.
It would be finite but unbounded.
That's what I'm referring to here.
- There is nothing outside of our universe. Nothing that is of interest for our physics, or any other science, for that matter. There's nothing for our understanding of "The Answer to the Great Question, of Life, the Universe and Everything". So, one can even say there's no outside at all.
- Einstein bent spacetime (Really. He did. Agreed, not only him. You do, too. Why? Well, he had mass. You have, too.). In short, mass bends spacetime. That means, even if you travel in uniform motion, the geodesic you travel on is almost never a straight line, because there's mass in the universe everywhere, much mass (There exist also masses of non-baryonic matter but I'm not going to consider those [here and now]. I agree, the universe is also big, really big, vastly, hugely, mindbogglingly big, so the average density of matter is low, very low, tinily, minusculely, ...[-1] low on a large scale (one proton in a cube with a 5 m edges, according to Prof. Bartelmann). However, gravity cannot be suspended, not even shielded, it always acts and always acts attractively (in multiple meanings of this word), it's simply there, everytime, everywhere (Great! That last even rhymes. Well done!). That means, unless the bending effect isn't smaller than one Planck length (that's 0.000000000000000000000000000000000016 m, BTW) there will be a tiny, minuscule, ... effect, at least, and it adds up.
Are these assumptions valid?
If a spaceman – let's call him Stephen – in a spacecraft – let's call it Hawk 42 – travels straight towards the border of the universe – and there is one, see 1. above – the closer he comes to it the more of the vast mass, nearly all mass at a certain point (apart from the spacecraft's cockpit instruments and its nose) is located behind him. In front of him pure nothingness, pure void, see 2. (I'm feeling a bit like on an esoteric trip with these words, but there aren't any better I'm aware of. Sorry for that.).
Now imagine what happens with space – that's bent by masses, see 3. – when all of the mass is located on one side of it and on the other side is simply nothing: it bends totally to the side with the masses whereby 'totally' means 180°. That means, space bends back into itself.
What happens to Stephen now? Nothing he recognizes immediately. He travels straight ahead like all the billions of light-years before, without even having the theoretical chance to recognize that he comes close to the border of the universe. He will recognize that there is a change of direction, first slightly then more and more until 180° if he looks at the star constellations that change in front of him – note: 'in front' becomes the former 'behind' more and more, automagically – until he's on his way back home without having to activate a single engine. (A very efficient way of travelling, isn't it?)
Key point is that from Stephen's point of view it seems as if he travels on a straight line while he's actually travelling on a universe-wide ellipse-like geodesic (A perfect ellipse if the universe would be a hollow sphere with its masses distributed equally, right?) due to space bent by the universe's masses. And this goes on forever since the same happens when he reaches the universe's border at the opposite side. So, for him it seems as if the universe were infinite, just the stars and planets repeat every few billion light-years – interestingly alternating: once from one side, once from the other side. Not later than that he should know... But hey! That's Stephen, he knew that before all of us.
Is this a reasonable explanation for the universe's endlessness? If it is not, is it a better metaphor, at least, than the poor, unfortunate ant on the sphere, crawling there for eternal times, without a chance to reach an end? I mean, compared to that Stephen is on his way home, repeatedly, after every few light-years.
1.↑ Douglas Adams: The Hitchhiker's Guide to the Galaxy, 1979