Hi, I don't think that the example of Gromov-Hausdorff is really an example. I mean, that is certainly a bounded metric space but is not compact. To see why consider the sequence where the $n$-th term is the space of $n$ points each with distance 1 from each other. This has no convergent subsequence in the sense of GH. 

A set of metric spaces si relatively compact w.r.t. GH if and only if is it uniformly totally bounded, i.e. for every $\epsilon$ there exists $N(\epsilon)$ such that every space in the set can be covered with at most $N(\epsilon)$ balls of radius $\epsilon$ (Gromov).

P.s. I would have liked to post this as a comment, but don't really know how to, first post here