Yet I never seem to catch 'em at the local pub.

For each finite dimension n (1, or 2, or 3, etc...), the sphere in dimension n can't be contracted because of that empty n-dimensional space it surrounds. But that same sphere is the "equator" of the sphere in the next higher dimension, n+1. There, the n-dimensional equator can contract along one of the hemispheres, to a pole. But then that whole (n+1)-dimensional sphere still isn't contractible, because of the (n+1)-dimensional space it surrounds.

BUT the (n+1)-dimensional sphere can contract along one of the hemispheres in the (n+2)-dimensional sphere. And so on.

For any particular finite dimension n, there is an n-dimensional obstruction to contracting the sphere in that dimension. But if you go all the way to infinitely-many dimensions, there is no obstruction that ever stops contractibility of the infinite-dimensional sphere.

Sphere refers to the surface area, correct? Ball would be the volume inside?

What is contractible in mathematics? Wikipedia had article on contractible spaces but that was way beyond be.

most people would imagine

I imagine that most people don't have the faintest clue what you're talking about, though

Wait, I thought the volume of a sphere approaches 0 as dimensions go to infinity, no? Thr general formula for the volume of any nth dimensional sphere has the gamma function in the denominator, which rises faster than whatever is in the numerator. At some point (5 dimensions, iirc) the volume starts decreasing