So I sat down and thoroughly calculated the Christoffel Symbols and Ricci tensor for the following 2-dimensional metric:
$$g_{xx}=R^2\frac{1-y^2}{(1-x^2-y^2)^2},\,g_{yy}=R^2\frac{1-x^2}{(1-x^2-y^2)^2},\,g_{xy}=g_{yx}=R^2\frac{xy}{(1-x^2-y^2)^2}$$
where $x^2+y^2<1$.
In the book I found this metric, it was mentioned that for this metric the distance between two points is unbounded. It turns out that the Ricci scalar is $R=-\frac{2}{R^2}$. So I am troubled with two things:
(1) Why is the distance between two objects unbounded when the space is described by that metric? (2) Is there any connection between the surface of a 2-d sphere and this particular metric?
OK, I've double-checked your computations (with assistance from Mathematica). The Gauss equations do give $K=1$ with $R=1$. I believe this is just an open-disk model for a hemisphere using a combination of orthogonal projection and stereographic projection. The surface is not complete, but certainly the distances stay bounded. (For example, check the metric is rotationally symmetric about the origin. Now compute the length of the line segment from $(0,0)$ to $(1-\epsilon,0)$ and you'll see this approaches $\pi/2$, as it should.)