Union of fat Cantor sets?

For $\mathcal{C}_k$, at the $n$th iteration you remove a middle interval of length $1/k^n$, so the complement of $\mathcal{C}_k$ has measure $$\sum_{n=0}^\infty \frac{2^n}{k^{n+1}}=\frac{1}{k-2}$$ Hence, when $k$ goes to infinity you have $m([0,1]\setminus\mathcal{C}_k)\rightarrow 0$ and hence it works.

Actually, I wonder, is there anything in the intersection of all the complements apart from the middle point $0.5$? Of course there is, as it is a comeager set hence a countable intersection of dense sets, as user254665 noticed.

Take open intervals around all points of $\mathbb{Q}$ that get smaller and smaller to define an open set $O_n$ of measure $\le \frac{1}{n}$. Then $D=\cap O_n$ has measure $0$ and is a dense $G_\delta$ so its complement is meagre.