In particular, the reals mean you accept measure theory, which in one of its key results says you can build two spheres of radius x from one sphere of radius x, and no holes. The computables don't go there.
There are infinitely many computable numbers, but only countably many -- no more than of natural or rational numbers. That turns out to be enough for everything sane you want to do.
Of course you need altered versions of the key theorems of calculus, because the computables are not continuous in the "real" sense. The numbers blur a bit, instead, to cover the gaps, much as water manages to behave like a fluid despite being made of nothing but discrete particles. The differences are a PITA but keep you on the straight and narrow. You get the same answers for everything that makes sense, and no answer for things that don't.
All the supposedly real numbers you will ever encounter are computable (too). Roots, pi, e, anything representable with a Taylor series. So you don't really give anything up.
Reals make a good enough approximation, which means you don't need to go hungry. Pretend you're using reals. Nobody needs to know. Everybody else is, too. The sane, anyway.