"Fat" Cantor Set
Say you delete the middle third.
Then delete two intervals the sum of whose lengths is $1/6$ from the two remaining intervals.
Then delete intervals the sum of whose lengths is $1/12$, one from each of the four remaining intervals.
And so on. The amount you delete is $\displaystyle\frac13+\frac16+\frac{1}{12}+\cdots= \frac23.$ That is less than the whole measure of the interval $[0,1]$ from which you're deleting things.