You are looking for the notion of Dieudonne complete spaces (those are exactly the closed subspaces of products of metric spaces). As Todd mentioned in his comment, this notion is closely related with the notion of realcompactness (the two notions coincide if there are no measurable cardinals). A way to find examples is to look for pseudo-compact non-compact spaces (for instance $\omega_1$ with the ordertopology).

You are looking for the notion of Dieudonne complete spaces (which turn out to be exactly the closed subspaces of products of metric spaces). As Todd mentioned in his comment, this notion is closely related with the notion of realcompactness (the two notions coincide if there are no measurable cardinals). A way to find examples is to look for pseudo-compact non-compact spaces (for instance $\omega_1$ with (a uniformity compatible with) the order topology).