By the Central Limit Theorem, something like this should have the
property that $0<\mu([a,b]\cap A)<b-a$ for all $0\leq a<b\leq 1$. Let
$0.a_1a_2\cdots$ be the binary expansion of $x\in[0,1]$. Let $A$
consist of all $x$ such that for all $n$ sufficiently large,
  $$ -1 < \frac{\frac n2-(a_1+a_2+\cdots+a_n)}{\sqrt{n}}<1. $$
The bounds $-1$ and $1$ could be replaced by any $\alpha<\beta$ in
$\mathbb{R}$.