Let K denote an inertial frame.
(Assumption) Particle trajectories are continuous in K. That is, the t ↦ (x, y, z) relation is a continuous function, where t denotes the coordinate time in K and (x, y, z) the momentary location of the particle relative to K.
Notes: (a) backed by the fact that the speed of light is finite, similar can be assumed about signals too, (b) the assumption implies that particle speeds are always finite.
(Definition) In K, the distance between two particles at coordinate time t is the distance between their respective momentary locations.
This definition is compatible with the classical conception of distance, and includes the possibility that the particles are moving relative to K. So we rely on coordinate time to define the distance between moving particles.
As long as there is only one inertial frame considered, space seems no different from that of classical mechanics.