The maximal ideal space $\Delta$ of a finitely generated Banach algebra is homeomorphic to a compact subset of $\mathbb{C}^n$. On the other hand, evaluation at each point of $X$ is clearly a complex homomorphism of $C(X)$. We conclude that $X$ is homeomorphic to a subset of a compact subset of $\mathbb{C}^n$.
Edit: actually, if $X$ is not compact $C(X)$ is not a Banach algebra at all (locally compact for $C_0(X)$). In locally compact case, as Nik Weaver showed in comments, $\Delta \cup \{0\}$ is compact in $\mathbb{C}^n$, so $\Delta$ is homeomorphic to a closed subset of some Euclidean space.
Also, Stone-Weierstrass theorem shows that all such $X$ satisfy the condition.