Translates of a set of positive Lebesgue measure cover $\mathbb{R}$?

Let $E$ be a set of positive Lebesgue measure in $\mathbb{R}$. Does some countable union of translates of $E$ cover $\mathbb{R}$?

My intuition is that $\mathbb{R}$ can be covered with countable translates of $E$ except a set of measure zero. Clearly, if $E$ contains an interval, its translates will cover $\mathbb{R}$. So the example that I tried to make work was the set of irrationals, but I don’t know how to show that countable union of its translates cannot cover $\mathbb{R}$.

Any hint or solution is appreciated.