Borel sets which are null or non-$\sigma$-finite for every translation invariant measure

We show that the set of Liouville numbers is either null or non-$\sigma$-finite with respect to every translation invariant Borel measure on $\text{RR}$, in particular, with respect to every Hausdorff measure ${\text{iH}}^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\text{RR}$ that are not $F_{\sigma }$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.