Is $L = \{ x \in \{ 0, 1 \}^* : |x| = 2^n $ for some natural number n $\}$ context free?
Suppose $L$ was context-free. Then $L \cap 0^*$ were also context-free. Since $L \cap 0^*$ is a unary language (a language over a unary alphabet, i.e., an alphabet of size $1$), Parikh's theorem states that it is context-free iff it is regular. As you mention, $L \cap 0^*$ isn't regular, and so $L$ can't be context-free.