Show that $L$ and $\overline L$ cannot be both finite
If $L$ and $\overline{L}$ were both finite, then so would $\Sigma^* = L \cup \overline{L}$ be, which we know is false.
This is a qualitative statement. We can make it quantitative by considering densities. Let us say that a language is dense if for infinitely many $n$, it contains at least half the words of length $n$. It is easy to check that at least one of $L,\overline{L}$ is dense.