This question was answered recently by Geschke, Grebík, and Miller:
S. Geschke, J. Grebík, and B. D. Miller, "Scrambled Cantor sets," Proceedings of the AMS 149 (link to the arxiv version)
They show that if $X$ is analytic (i.e., the continuous image of a Polish space), and if $(X,f)$ is a dynamical system with Li-Yorke chaos, then $(X,f)$ contains a scrambled Cantor set.
Let me point out that some assumption on $X$ (such as "analytic") is necessary here. For example, this is easiest to see if CH fails. Why? Let $(Y,f)$ be any dynamical system with Li-Yorke chaos. This means that it contains a scrambled set of size $\aleph_1$. Let $X$ denote the closure of this set under $f$, and observe that $(X,f)$ is then a dynamical system with Li-Yorke chaos, but its cardinality is smaller than that of the Cantor space! If CH holds, one can construct such things via transfinite recursion.