Covering null sets by a finite number of intervals

The set $\mathbb Z$ has measure $0$ and it is nowhere dense. However, the property $P$ doesn't hold for $\mathbb Z$.

For an example of a bounded nowhere dense set of measure zero which does not have your property $P$, let $F$ be a compact nowhere dense set of positive measure (a fat Cantor set), and let $A$ be a countable dense subset of $F$.

In fact, it's easy to see that a set $A\subseteq\mathbb R$ has property $P$ if and only the closure of $A$ is a compact set of measure zero.