If you allow $A$ to contain an interval, let
$$A = (\mathbb Q\cap [0,1])\cup [0,1/2].$$
If not,
consider $q_i$ the $i$th rational number in some enumeration of $\mathbb Q\cap [0,1]$, and let
$$A=\bigcup^\infty_{i=1} (q_i-4^{-i}, q_i+4^{-i})$$