Open set containing rationals but complement non-denumerable

The sum of the measures of these sets is $\sum\limits_{n=1}^\infty \dfrac 1 {2^{n-1}} = 2$. Consequently the measure of its complement is $\infty$. That the measure of the complement is positive is enough to entail that the complement is uncountable.

For a non measure-theoretic proof you can construct a Cantor set containing no rationals and then take its complement. Here's a sketch of how to make it:

Start with an interval $[a,b]$ with $a,b$ irrational. Let $\{q_n\}$ be an enumeration of the rationals in this interval. Now remove from $[a,b]$ an interval $(c,d)$ with irrational endpoints containing $q_1$. Now remove open intervals with irrational endpoints from the two remaining closed intervals so that $q_2$ is no longer in the set, and so on.

The result is a closed set avoiding every rational number which has cardinality equal to that of the continuum (equivalently, to the set of all infinite binary sequences). The proof that Cantor sets have cardinality continuum is very similar to the proof that the standard Cantor ternary set consists of all numbers with only 0 and 2 in their ternary expansion (Or since its a perfect set the Baire category theorem shows that it's uncountable).