Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have this property. Using iterated function systems it is easy to construct further examples.

I wonder I there is a quantity that distinguishes the "size" of such sets. The quantity should be a least monotone with respect to inclusion and stable with respect to unions.