Uncountable sets of Hausdorff dimension zero
A thin version of the usual Cantor middle third set works. The idea is just that you need to omit more than just a third of the remaining intervals as the construction proceeds, enough so as to force the Hausdorff dimension to $0$.
Specifically, we construct the set in stages. At each stage, we've omitted a "middle third" from each finite interval remaining. Thus, at stage $n$, our set is contained in $2^n$ many intervals of some finite length $a_n$. In the typical middle-third construction, we have $a_n=3^{-n}$. But in our construction here, we want $a_n$ to be small enough that $2^na_n^{1/n}\to 0$. By this means, the Hausdorff dimension will be forced to $0$. But the resulting set is perfect, and hence is uncountable of size continuum.