Is decomposing the middle third Cantor set doable paradoxically? I was just curious, is it possible to break the middle third Cantor set $C$ into a finite number of pieces, and after rearranging them, obtaining a "larger" set, i.e. perhaps in terms of Lebesgue measure? Has this even been attempted as yet?

It is not possible as asked, because the Cantor set has measure zero; _every_ subset of a measure zero set is Lebesgue measurable with measure zero; the "rearranging" operations would preserve measure; and a finite or countable union of measure zero sets still has measure zero.