The authors study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system of the type \[ \begin{cases} -\Delta u=f_\lambda(x)|u|^{q-2}u+{\alpha\over{\alpha+\beta}}h(x)|u|^{\alpha-2}u|v|^\beta &\text{in}\;\Omega,\\ -\Delta v=g_\mu(x)|v|^{q-2}v+{\beta\over{\alpha+\beta}}h(x)|u|^{\alpha}|v|^{\beta-2}v &\text{in}\;\Omega,\\ u=v=0 &\text{on}\;\partial\Omega, \end{cases} \] involving critical exponents and sign-changing weight functions. In particular, \(\Omega\subset\mathbb R^N\) is a bounded domain, \(N\geq3\), \(0\in\Omega\), \(\alpha\), \(\beta>1\), \(\alpha+\beta=2^\ast ={{2N}\over{N-2}}\), \(q\in(1,2),\) \(\lambda\), \(\mu\geq 0\). Using the Nehari manifold, the authors prove that the system have at least two nontrivial nonnegative solutions when the pair of the parameters \((\lambda,\mu)\) belongs to a certain subset of \(\mathbb R^2\).