Why is this basic language not a regular language?
Every language is the union of regular languages:
$$
L = \bigcup_{x \in L} \{ x \}.
$$
However, the set of regular languages is not closed under infinite unions. It isn't even closed under countably infinite unions, as your example demonstrates.

You can prove that your language is not regular in many ways. For example, if your language were regular, then so would its intersection with $0^*1^*$ be; yet this language is $\{ 0^n 1^n : n \geq 0 \}$, the classical example of a non-regular language. You can also prove non-regularity of your language directly, using either the pumping lemma or Myhill–Nerode theory.