TM recognizing $0^n1^n$ requires Ω(log n) space
As you correctly point out by counting configurations, it holds that $SPACE[o(\log n)]\subseteq TIME[o(n)]$.

There is a theorem stating that $TIME[o(n\log n)]=REG$. See e.g. here.

If, by contradiction, we had $\{0^n1^n: n\in \mathbb{N}\}\in SPACE[o(\log n)]$, it would then follow that $\{0^n1^n: n\in \mathbb{N}\}\in REG$, which is clearly false.

Thus, $\{0^n1^n: n\in \mathbb{N}\}$ cannot be solved in $SPACE[o(\log n)]$.