I used to be really confused by this until I really looked at the definitions of closed and open sets.

By definition, a set $S$ is closed if $S^c$ is open, and vice-versa.

The empty set $\emptyset$ trivially has nothing but interior points, and also trivially includes its boundary. This took me a while to internalize, but I think it's easiest to just ¯\_(ツ)_/¯ and accept it.

Since $\emptyset$ is open, then $\emptyset^c = \mathbb{C}$ is closed. And since $\emptyset$ is closed, then $\emptyset^c = \mathbb{C}$ is open.

Therefore, $\mathbb{C}$ is both open and closed.