Solution:
Let $d$ be the expected value of the absolute difference. Observe that the Cantor set is made up of two smaller copies of itself, each scaled down by a factor of $3$. There is a $\frac{1}{2}$ chance that the two selected numbers are in the same copy, in which case the expected value of their absolute difference is $\frac{1}{3} d$. Otherwise, we can write them as $\frac{2 + x}{3}$ and $\frac{y}{3}$ for independently and uniformly randomly selected $x$ and $y$ in the Cantor set. Their difference is $\frac{2 + (x - y)}{3}$, which by symmetry has expected value $\frac{2}{3}$. Thus
$$d = \frac{1}{2}\cdot \frac{1}{3} d + \frac{1}{2}\cdot \frac{2}{3} \Rightarrow d = \left[\frac{2}{5}\right].$$