This question already has an answer here:
This might be silly, but I am not sure:
Does there exist a Lebesgue measurable subset $E \subseteq (0,1)$ such that
$E$ and $(0,1) \setminus E$ both have positive Lebesgue measure.
$E$ and $(0,1) \setminus E$ both have empty interiors.
If we relax condition $1$, then $E=Q\cap (0,1)$ works. If we relax condition $2$, then the fat Cantor set does the job. (Its complement have non-empty interior though).