Space does indeed have structure. There are three structure that are immediately visible when we think of it as a manifold.
- Continuous structure
- Smooth structure
- Metric structure
The first says that curves in space are continuous; the second says that they have no kinks; and the third says we have a distance function between points.
They are in order of increasing strictness; for example, a metric structure determines a smooth and continuous structure.
Since relativity up-ended our notions of space and time we ought not to speak of space and time separately but together - spacetime. The above still holds but now we can add a time orientation. One might think that here we have an additional structure - causality. But this in fact is implicit in the metric structure.
And also since QM up-ended our notions of what physics means at small distances there has been a further revolution in our understanding of space. Contemporary thinking supposes space has a discrete structure.
For example, in Loop Quantum Gravity, the area and volume operators have a discrete spectrum; this means that area and volume comes in discretely sized blocks. This is taken as a starting point in causal set theory where spacetime evolves in discrete jumps.
There is a long history attached to the conceptualisation of space. Even as far back as Aristotle, this is two and a half millenia ago he pointed out that space was a thing, it was a place which could be occupied by a thing; he pointed this out to dismiss the atomists conceptualisation of space as a void; for him, there was no such thing as void, and he argued that the atomists void was better conceptualised as place.