Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth open set.We prove that the singular set of any extremal solution of the system\begin{equation*}-\Delta u=\mu e^v , \quad- \Delta v=\lambda e^u\quad\mbox{ in }\Omega,\end{equation*}with $u=v=0$ on $\partial \Omega$, $\mu,\lambda\geq0$, has Hausdorff dimension at most $n-10$.