Marc Rieffel has studied a notion of `quantum' compact metric space that has its roots in the picture of unital C*-algebras as noncommutative compact Hausdorff spaces: see http://arxiv.org/pdf/math/9906151v2 and http://arxiv.org/abs/math/0011063, for example. One should consider this answer as a supplement to the answers above citing the work of Nik Weaver and Greg Kuperberg.
Note: Rieffel uses the word 'quantum' in place of 'noncommutative' because the multiplicative structure doesn't really play a role.