Let $(r_k)_{k \in \mathbb N}$ be an enumeration of $\mathbb Q$
and $U_{1/p} := \cup_{k \in \mathbb N}(r_k - \frac{1/p}{2^k}, r_k + \frac{1/p}{2^k}) \subset \mathbb R$
What is $\cap_{p \in\mathbb N^*} U_{1/p}$ ?
It seems it cannot be $\mathbb R$ (because the measure of the sets is finite) or $\mathbb Q$ (because it can't be a countable intersection of dense open sets), so I don't see...
Note that the Lebesgue measure of $U_{1/p}$ is, at most$$\frac2p\sum_{k=0}^\infty\frac1{2^k}=\frac4p.$$Therefore, the Lebesgue measure of $\bigcap_{p\in\mathbb{N}^*}U_\frac1p$ is $0$. So, $\bigcap_{p\in\mathbb{N}^*}U_\frac1p$ is an example of a dense subset of $\mathbb R$ with Lebesgue measure $0$; in particular, it is not $\mathbb R$. But it is not $\mathbb Q$ either, because it is not countable. And it is not countable because it follows from that Baire category theorem that no countable dense subset of $\mathbb R$ can be obtained as the intersection of a countable family of open sets.