Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$.
Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty \;I_{ij},\;\forall\;j\in\{1,2,3,\dots\}$.
Finally, let $B = \cap_{j=1}^\infty \; G_j$.
Then it follows from a theorem of Borel, and from the construction above, that $B \supseteq \mathbb{Q}$ is a nullset.
What's in $B\,\backslash \mathbb{Q}\,$?
Is possible to exhibit any elements of $B\,\backslash \mathbb{Q}\,$ ?
Is possible to characterize the elements of $B\,\backslash \mathbb{Q}\,$ any further?