Subsets $X$ such that their Hausdorff outer measure is not finite

Yes, if $X$ is an analytic set. See:

• Besicovitch, On existence of subsets of finite measure of sets of infinite measure, 1952
• R. O. Davies, Subsets of finite measure in analytic sets, 1951.

For discussion you may consult e.g.

http://arxiv.org/abs/1408.1999