Vitali set $V$:

https://en.wikipedia.org/wiki/Vitali_set

Cantor ternary set $\mathcal{C}$:

https://en.wikipedia.org/wiki/Cantor_set

Equivalently, is the following statement true?

$$\exists V \in \{y : y \text{ is a Vitali set}\}(x \in V \implies x \in \mathcal{C})$$

I don't believe there is a Vitali set that is a subset of $\mathcal{C}$ because the Lebesgue Measure of $\mathcal{C}$ is $0$ while the measure of a Vitali set is undefined (if not positive in the sense that the union of countably many Vitali sets may have a positive real measure so it's possible to consider the measure of a single Vitali set as being both positive and infinitesimal in addition to being non-Lebesgue measurable).