A set like the rationals has Lebesgue measure 0 and its closure are the real numbers which has positive Lebesgue measure. However Q is dense. If a set where to be nowhere dense and also null what would the measure of its closure be in R?
A fat Cantor set, a small modification of the construction of the middle-third Cantor set in $[0,1]$ is nowhere dense and compact and can be given any measure $m>0$. A countable dense set $D$ of this fact Cantor set is then a nowhere dense null-set of measure $m$.
Nowhere dense and null-set are unrelated. (But in the reals a closed null set is nowhere dense).