this is basically a question on the granularity of space, which is an active topic of research in physics: space appears to be continuous, but does it actually come in discrete chunks on some very small length scale (Planck length)? there are some attempts to formulate (quantum) mechanics in discrete space-time; loop-quantum-gravity is one approach, described here by Lee Smolin; a alternative approach is promoted by Gerard 't Hooft:

*In modern science, real numbers play such a fundamental role that it is difficult to imag- ine a world without real numbers. Nevertheless, one may suspect that real numbers are nothing but a human invention. By chance, humanity discovered over 2000 years ago that our world can be understood very accurately if we phraze its laws and its symmetries by manipulating real numbers, not only using addition and multiplication, but also subtraction and division, and later of course also the extremely rich mathematical machinery beyond that, manipulations that do not work so well for integers alone, or even more limited quantities such as Boolean variables.
Now imagine that, in contrast to these appearances, the real world, at its most fundamental level, were not based on real numbers at all. We here consider systems where only the integers describe what happens at a deeper level. Can one understand why our world appears to be based on real numbers?*