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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 $\mathbb{R}$.