The inside of a sphere. http://i.imgur.com/b2KgM6X.gifv

That was really interesting, do you have a source? (The title makes it seem like a joke though)

Google Hollow Earth. There are many conspiracy theory sites handling it, but I think it is a quite interesting concept nontheless.

No, paraboloid also has positive curvature. Sign of curvature refers to whether all curves at a point are in the same half-space (saddles have negative curvature).

OP really means something like "eccentricity <1"