Imagine the universe is made out of graph paper, with only the points of intersection of the lines being actually "real". That is, space is actually discrete. If that is true, then you cannot meaningfully talk about things being "half a square" apart.
People (some people) think that the universe really is that way, at the Planck distance. Actual experimental confirmation is somewhat lacking at this time...
More importantly, this idea doesn't work in the context of space dilation. Per special relativity, the distance between objects is arbitrarily different for different observers, so any grid you draw will be wrong for me if I'm moving at a different velocity than you. This is especially bad because it doesn't just depend on the magnitude of the velocity, but also the direction - space is compressed in the direction of motion. So two observers moving at an angle to each other will have very different views of the grid.