I think for maximum uselessness, they should not be overlapping spheres, but deform at the interface, like soap bubbles or rubber balls. As long as the spheres are the same size and modelled with the same "surface tension" or "elasticity", the "intersection" of two sets would then be a circular interface with an area proportional to what would otherwise be an overlap (I think). If the spheres have different sizes or are modelled with different surface tension or elasticity, one would "intrude" into the other.

Multiple sets would have increasingly complex shapes that may or not also create volumes external to the deformed spheres but still surrounded by the various interfaces.

Time to break out the mathematics of bubbles and foam. This data ain't gonna obscure itself!

Might there actually be utility to something like this? Scrunch the spheres together but make invisible everything that is not an interface and label the faces accordingly. I suppose the same could be said of the shape described by overlapping. (Jesus, you'd think I was high or something. Just riffing.)

Venn diagrams, but the sets represent whatever the diagram is about (like houses for housing markets).

Projected on a 2D screen, it'd look more like a normal venn diagram.

Volumetric Herbert space diagram.

Why limit it to 3 dimensions?

How about 4D Venn diagrams?