Lemma: For any enumeration of binary sequences, there exists a binary sequence not in that enumeration.

Proof: Let $L$ be any enumeration of binary sequences. Let $x$ be a binary sequence such that $x_i \neq L_{i,i}$. Then $x \neq L_i$ for any $i$. Thus $x$ is not in $L$.

Theorem: There is no enumeration of all binary sequences.

Suppose there is an enumeration of all binary sequences. By our lemma, there is a binary sequence not in this enumeration. But this contradicts the assumption the enumeration contains all binary sequences. Contradiction. Thus no such enumeration exists.