The Wikipedia article on Hyperbolic Geometry has a simple image of a triangle in a “saddle” surface. In 3D, a “cube” will have a warped, melted look like a Dali painting.
You can construct something with a lot of such saddle surfaces instead of semi-spheres. Walking on such a surface will be fun.
If you have a handy little black hole nearby you have some non-Euclidean space to build in.
(Technically, the gravity distortion of the Earth itself will do, but it won’t be obvious to the naked eye.)