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 nonmeasurable. You can prove that all the measurable sets of positive measure have the same cardinality as the Continuum. In case that the set is nonmeasurable it's outer measure would be positive. 

