[set fair (OP)]

If the space where we live is positively curved by gravity and the nearly empty deep space is negatively curved then this suggests that there is a particular density of unit mass per unit vilume which would be flat. Are there estimates of what this density is, It might turn out to be an illuminating value.

[Halc]

If the space where we live is positively curved by gravity

Gravity curves spacetime, not space.
Spacetime is negatively curved by mass, meaning the angles of a triangle defined anywhere will add up to less than 180 degrees.

this suggests that there is a particular density of unit mass per unit vilume which would be flat

A completely even distribution of mass would be flat in Minkowski spacetime, but on the largest scales, spacetime is not Minkowskian.

On a more local level, spacetime is flat inside a uniform hollow sphere. Is this what you're imagining?

There's no mass within that spacetime, so it's mass-density is zero. You're not being clear exactly what you mean by 'dense'?  Density is an amount of something per unit <something else>, but you've not indicated what the 'something' is.

[set fair (OP)]

On a more local level, spacetime is flat inside a uniform hollow sphere. Is this what you're imagining?

Thanks - what I'm imagining is: if you set out from earth where spacetime is positively curved and voyaged out to deep space where the curvature is negative wouldn't you pass through a space or spacetime which is flat?

[Halc]

Spacetime (again, not space) is not necessarily positively curved if you set out from Earth.
It may be positively curved on the grandest cosmological scale, but such curvature has never been measured.

Yes, spacetime might approach approximate flatness in the areas between superclusters.

There are large regions of local positive curvature in areas of 'negative gravity'* such as the dipole repeller (a more concentrated curved repeller) or the cold-spot repeller, which is larger but less curved due to being more spread out.
There are smaller locations of local potential maximums which are positively curved, such as between stars.
Spacetime is locally flat anywhere, and only takes on curvature as the scale is increased.  So reasonably flat spacetime can be momentarily experienced simply by jumping off a cliff.

* Negative gravity means areas of higher gravitational potential than the surrounding areas in any direction.  Matter is repelled from such regions due to the lower potential in every direction.  Higher potential is a relative term.  All gravitational potential is negative since somewhere there is a hypothetical zero beyond which there is no more potential.  This is the equivalent potential of a universe devoid of all matter/energy.