This is the (a) Vitali set, which is not Lebesgue measurable. A quick reason is: 

$\mathbb{R}=  (v_0\mathbb{Q})\oplus_\mathbb{Q} V$ shows that $\mathbb{R}$ is a countable union of translates of $V$, so $V$ cannot be a Lebesgue set of measure zero, but must have positive outer measure. By translation invariance and by the Caratheodory criterion for intervals, it follows that in fact $V$ has infinite outer measure.

On the other hand, for any Lebesgue measurable set of positive measure $S$, according to  Steinhaus property, $S-S$ is a nbd of $0$. Therefore $V$, which is nowhere dense, contains no set of positive measure.