Given any $\epsilon > 0$ and given any nowhere dense set $N$ of positive measure, there exists a Cantor set $C$ (Cantor set here meaning a perfect nowhere dense set) such that $C \subseteq N$ and the measure of $N - C$ is less than $\epsilon.$ This is essentially a result that, in real analysis texts, is often called Lusin's theorem. It's usually stated with $C$ being a closed set, but the Cantor-Bendixson theorem says you can turn any closed set into a perfect set by deleting at most countably many selected points from the closed set, and this is a process that doesn't affect the Lebesgue measure. So, once you get the nowhere dense closed set that the usual Lusin's theorem gives you (it'll be nowhere dense because it's a subset of $N,$ and we're assuming $N$ is nowhere dense), you can toss out at most countably many points and get a Cantor set that has the same measure.
Thus, any nowhere dense set of positive measure is, except for a left-over part having arbitrarily small measure, a Cantor set of positive measure. Another way to view this is that you can get all nowhere dense sets of positive measure by arbitrarily small (in the sense of Lebesgue measure) enlargments of Cantor sets of positive measure.