Very nice question:)

When you want to define something you can do it two ways. By making what you define very small in a flat SpaceTime, like on a table. Then there is no 'gravitation' (theoretically) interfering with what you want to define. That also assumes that what you define is infinitely small in fact, and that 'gravity' can't get a 'hold' on it.

Or you look at SpaceTime as you see it a dark night, it all being there, in a equilibrium. Then gravity is 'everywhere', no place excluded as I know, and no 'point' too small. In SpaceTime gravity is expressed through the stress energy tensor which is a combination of density, invariant mass, momentum, pressure and the stress in matter that all combines into the warping of the SpaceTime fabric. To have a 'space' you will need objects defining it. Without any objects, how will you decide that there is a space? Space is a relation between those objects that we measure calling their relation a distance. The distance we define comes from using clocks and some type of ruler. That is, we need a clock and then we need to define a unit for measuring with, I can't see any simpler way. I'm not sure if it would be enough with only one object in a space to define it. To get a distance I think you at least will need two.

So does space 'grow' away from mass?

I would expect it to be constant myself, assuming that 'gravity' is everywhere. That as you can see 'gravity' also as something communicating the SpaceTime we live in. You might also exchange 'gravity' for light, and then decide that as long as light can reach you between those empty stretches of SpaceTime, then SpaceTime has to 'know' its distances. All of this comes from our discussion on how many objects you need to define a 'space'. But there are other ways to define it, as redshift. And there we have phenomena we don't really understand even though we can deduce that they seem to happen.