Properties of convex polytope of 0-1 matrices
Sasho already gave you a yes/no answer, but here's an actual convex combination for you: If $B_\ell$ is the $k \times k$ matrix which is $1$ when $|S_i \cap S_j| \ge \ell$ and zero otherwise, then $M = \sum_{\ell=1}^{n'} \frac{1}{n'} B_\ell$.