The standard view of spacetime is that it's a manifold based on the real numbers. The view that spacetime is discrete is also sometimes considered. Much more rarely, the idea that spacetime is a "manifold" based on a field like the hyperreals that extends the reals gets considered. But I have never heard anybody wonder whether spacetime might not have a dense but countable structure, like the structure of the rational or algebraic numbers.
I see no philosophical benefits to thinking spacetime might have such a structure. I wonder, though, whether we have any philosophical or empirical reason to think it doesn't. Perhaps there is a consistent first-order axiomatization of the mathematics that physicists actually use. If so, then the downward Loewenheim-Skolem theorem will model it within a countable structure. The algebraic numbers are unlikely to be rich enough, though.