What is the relationship between the Hausdorff dimension and cardinality of a set?

Specifically, assuming the Continuum Hypothesis, if a set has Hausdorff dimension greater than zero does, that imply that its cardinality is equal too or greater than that of $2^{\aleph_0}$?

Or, does the negation of CH, imply the existence of a set with positive Hausdorff dimension and cardinality strictly between $\aleph_0$ and $2^{\aleph_0}$?