If you look at an atom, its occupied space, is mostly filled in by the electrons. This atomic space is often modeled by probability functions of where the electron is, with the electron never being 100% in any given place, at a time; wave and particle. This suggests there is empty space, but in the case of an atom, all that spaces is occupied and not occupied a fraction of the time. That is only for one atom. It we have two atoms of a gas; helium, the space between atoms is even less occupied, but since the probability function still applies it is sometimes occupied, albeit not often.
If matter permeated all of space, you would have an effect similar to omnipresence. This effect would not be part of space-time, but more of an artifact of separated space and separated time, with time=0. If we factor out time, then all probability functions; position in space and time, does not apply. This may be possible at the quantum level, where separated space and separated time would apply.
Say we could travel at the speed of light; Special Relativity, our space-time universe would appear to contract to a point-instant. In this scenario, the matter of the universe will appear to overlap that point in space-time, thereby totally filling space; type of omnipresence. If we were to slow, from the speed of light, space-time will begin to expand making the universe and matter appear to expand, with time or probability gaps, forming within space-time.