Has anyone studied "polynomially compact" metric spaces?
The notion of bounded doubling dimension appears to be related. Specifically, assume your region S is a ball of radius \rho in the metric space. Then the space is said to have bounded doubling dimension $k$ if any such S can be covered by at most $(1/\epsilon)^k$ balls of radius $\epsilon \rho$. The difference between your definition and this one is the limitation of $S$ to be a ball though. 

In essence, your parameter $k$ is acting like a kind of dimension of the space. A worthwhile reference is Ken Clarkson's review of different notions of metric space dimension. 

Separately, I'm a little confused about your definition, since it appears to require a bound unrelated to the size of $S$. This is odd, because $S$ could grow arbitrarily large.