What are the surfaces of constant Gaussian curvature $K > 0$? Besides the sphere, is there any other surface with constant and positive Gaussian Curvature $K$?
The word "surface" is too vague. If it means a smooth subset of $\mathbb{R}^3$ that can be parametrized by two varibles it will "look" like the sphere of radius $1/\sqrt{K}$ near any point (will be locally isometric to a sphere in technical language). That could be any open piece of the sphere, or a disjoint collection of them. 

If we additionally assume that this subset must be closed in $\mathbb{R}^3$ then the entire sphere is the only possibility, up to moving it around. If we still want it closed, but allow it to be a subset of $\mathbb{R}^n$ with large enough $n$, then there is a second possibility, the real projective plane. It still "looks" like the sphere near any point, but has a different overall shape. 

The most abstract interpretation is to treat "surface" as not sitting inside any space (two-dimensional manifold), but still having some residual structure (called Riemannian metric), that makes the notion of curvature meaningful. However, even this does not lead to anything other than disjoint open pieces of spheres and real projective planes.
sieverts surface is a second example besides spherical surface. I guess that you want this answer.