Let $\{X_\alpha\}_{\alpha \in A}$ be a family of topological spaces with $|A| \ge 2$, and let each $X_\alpha$ be nonempty. The product $\prod_{\alpha \in A} X_\alpha$ (with the product topology) is locally compact if and only if each $X_\alpha$ is locally compact and all but finitely many of the $X_\alpha$ are compact.