Interesting question. The idea of the infinite contained within the finite has always fascinated me, and there are places where this is actually the case. There are mathematical objects known as fractals that are "infinitely complex," meaning that no matter how far you zoom in on them or how far you zoom out, and no matter how big or small a section of the fractal you focus on, you will always find rich and intricate detail. Yet the fractal appears to occupy finite space (look up fractals on google images, particularly the Koch Snowflake--they're very pretty to look at).
Another example is a coastline. Coastlines are fractals in nature (there are LOTS of fractals in nature, actually), and even though England is finite in size its shoreline is essentially infinite in length. The way to visualize is like this: say you want to measure the length of the shoreline. You take a meterstick and start measuring, one meter at a time. You go all the way around the island and you get a value. But as you go, you notice that there are parts of the shoreline that are smaller than a meter, and because the meterstick is straight and can't curve you weren't able to include those smaller dips and divots in the value you got. So you decide to try again, this time using a smaller ruler--say, a 10 cm ruler. You're able to measure more accurately since you can measure more of the smaller dips and divots, and by the time you finish you notice that your value is longer than you got with the meterstick, because you were able to measure bits that you had to skip with the larger unit of measurement. But even with the 10 cm stick, you still notice that you are missing even smaller dips and divots. This will continue forever. Eventually you could measure the lengths of the individual grains of sand that comprise the coast. Eventually you could measure the molecular and atomic contours of the particles that comprise the sand. Eventually you end up at the quantum level. But each time you use a smaller unit of measurement and are therefore able to measure the length of the shore more accurately, that length gets longer. We don't know just how small things can get in the universe. Perhaps there is a limit. But if there isn't a limit (and with fractals there is no limit to how small or large the fractal can be), then the units you use to measure can get infinitely small, and therefore the shoreline ends up being infinitely long. Yet we can see the shoreline, and it appears to inhabit finite space.
