Can the complement of a subset of $[0,1]$ that has $>0$ measure be dense in $[0,1]$

Question

Suppose $C\subset[0,1]$ has measure $\epsilon>0$. Can $[0,1]\backslash C$ be dense in $[0,1]$?

I'm thinking of a Cantor like set, where the construction of the set is a series of removals of $2^{k-1}$ intervals of length $\epsilon\cdot 2^{-2k+1}$