let's assume $\neg CH$, then there's a set $X$ such that $|\mathbb N|<|X|<|\mathbb R|$. i'm wondering about the lebesgue measure of such set... is it even possible to measure it? would it be possible that its measure is more than $0$? i think not, because all the subset of cantor set are measure $0$ sets, and there would be a set $K$ such that $|X|=|K|$ and $K\subset Cantor$. is it correct?
That set only have two options: it would be measurable (of $0$ measure) or it will be non-measurable. You can prove that all the measurable sets of positive measure have the same cardinality as the Continuum. In case that the set is non-measurable it's outer measure would be positive.