Is this a reasonable explanation for "the universe is finite but unbounded" which seems to be contradictory at first sight? [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"[1]. 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[1], 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
 A: Just a long comment for somebody who wants to visualize the universe as observations have defined it  for us, using the General Theory of Relativity : 

This is not an ant on a balloon, this is a two dimensional cut, representation, one space, one time, of our four dimensional universe (three space, one time), which mainstream physics accepts as the  standard model.
In the first simplistic Big Bang model, all the universe started at one point and expanded from there, in a four dimensional explosion.
We, and your Stephen, are always at the rightmost point in this graphic representaiton, 13,8+ billion years from that explosion from the original point. In this model this means that right now, all of us including stars galaxies and clusters of galaxies, are at the border of the universe, as it expands. The "unknown" border  is at the next instant of time, not of space. Space extends up to the curves shown in the graphic, and in the present cosmological mainstream model there is nothing outside those hyperbolic type lines. It cannot be reached in the sense of travelling there. Another way of seeing this is that one crosses the space borders every time one moves. Each of us is at a border given by the GR equations of the model.
In other words, for each of us, the (x, y, z) points we occupy were at the fuzzy point of discontinuity at beginning (t ~0) of the universe. Each of us is at the center of the universe.
Astrophysical arguments and observations forced a fuzzy, quantum mechanical  region about where the big bang point was in the simple model, and a rapid inflation period, followed by a slower inflation up to now. The enormous energy of the universe, when the spatial dimensions were smaller forced the creation of matter as the graph explains.
As time goes inexorably on, the distances between galaxies get larger, and the eventual fate of this universe is expected to be very diffuse photons and maybe neutrinos. The answer by Javier to the model of Gerold describes other possible models. The above is the mainstream one for cosmology at present.
So the universe you assign to Stephen is not a model of the universe we live in.
Addendum by Gerold Broser
There are two further illustrations:


*

*Nature Research: Box: Timeline of the inflationary Universe, nature.com, April 15, 2009



(source: nature.com) 


*

*Kavli Institute for Particle Astrophysics and Cosmology (KIPAC): Inflation, Stanford University, July 31, 2012



A: Assumption 1 is wrong, 2 is kind of philosophical (though more or less right), and 3 is correct. And every time I say this I feel bad, but I have to say it anyway: General Relativity is hard. People spend years learning all the background knowledge necessary to even begin to tackle learning GR itself. 99% of the time, coming up with conclusions without having formally studied the subject will lead to errors.
First let me explain why your argument is wrong, and then I'll try to talk about the ant on the sphere analogy.
Why is assumption 1 wrong? You're assuming the universe was finite at some point, but this is not true. See Did the Big Bang happen at a point? for an explanation. As far as we can tell today, the universe is infinite and has always been. You are probably correct in saying that if it was finite before it couldn't become infinite (though I'd have to do the math to be sure), but that's not the case (again, as far as we can tell).
I would just justify assumption 2 by saying that the universe is by definition everything that exists, so it makes no sense to say that something could be outside of it.
Finally, it's pretty nontrivial to figure out what kind of curvature a given distribution of matter will produce. If you were standing next to some kind of border it's true that all the matter would be behind you, but most of it would also be very far away, so I don't know if it would have a big effect. Also, your description of space curving back on itself is very vague; I for one don't know how to translate it into math to check if it makes sense.

The observations collected so far seem to say that the universe is infinite and more or less flat. Some years ago the most popular theory was that it was finite but borderless, like the sphere of the analogy. It's really hard to explain how this works in three dimensions without math, which is why people turn to the ant analogy. This analogy, by the way, is just that: an analogy. Of course it leaves out a whole dimension, because if we knew how to picture three dimensional curvature we wouldn't need the analogy. Unfortunately I think your best bet is just to trust the scientists who say that it makes sense. If you want to learn more to understand exactly how it goes, well, you have a few years of math ahead of you.
If you really want to try to visualize a periodic universe, try this one. Imagine the universe was the inside of a ball. We're 3D now, so no one is hiding any dimensions. This ball has a border, except it's not really a border. You should think of the whole sphere (the border of the ball) as being just one point: if you travel from the center to the "border" you can come out anywhere on this "border" depending on your direction of travel. In particular, if you go straight at it you will come out on the other side and eventually go back to the center. This is one way of visualizing a 3-sphere.
