Well, how can you go on forever without falling off the edge of the Earth? The surface of the Earth two-dimensional, and approximately flat... but there's small positive curvature, so you're continuously wrapping around to a different part of the space. The surface of the Earth can be meaningfully understood as a two-dimensional surface that possesses finite dimension.
A three-dimensional space can be close to flat but have positive curvature as well (or negative curvature, for that matter). Some proposals give our universe positive curvature, rendering its space finite, though still stupidly-big. (I'm not aware of there being a final word on the subject, though). And if different dimensions can have different curvatures, some of them could be much smaller than others.
My understanding is that we have conducted experiments to measure the curvature of the universe, and come up with answers that are withing the experimental error of being flat. Without any theoretical reason why the universe should be flat, it is still possible that the universe is curved, but to slightly for us to detect, however, but the simplest interpretation is to say that the universe is flat.
Of course, if we a assume a multiverse with universes having regularly distributed, positive curvatures, then, the size of the universe would grow asymptotically as the curvature approached flat, so, statistically, life is more more likely to arise on the flatter universes.
Is it possible, or in any way related, that a small but non-zero curvature could be responsible for the accelerated expansion of the universe? I don't know, it's hard to picture. But I'm thinking the idea that you can look at a sphere as a 2 dimensional space that curves and eventually wraps around such that things going opposite directions on its surface eventually can run into each other again.
That would probably be more easily measured though. I don't know.
A small non-zero curvature would correspond to a non-zero cosmological constant, which is precisely what seems to be driving the accelerated expansion of the universe (there are other explanations but this is the simplest one).
A three-dimensional space can be close to flat but have positive curvature as well (or negative curvature, for that matter). Some proposals give our universe positive curvature, rendering its space finite, though still stupidly-big. (I'm not aware of there being a final word on the subject, though). And if different dimensions can have different curvatures, some of them could be much smaller than others.