  Let $\mathcal{N}_g$ be the mapping class group of a non-orientable closed
surface. We prove that the proper cohomological dimension, the proper geometric
dimension, and the virtual cohomological dimension of $\mathcal{N}_g$ are equal
whenever $g\neq 4,5$. In particular, there exists a model for the classifying
space of $\mathcal{N}_g$ for proper actions of dimension
$\mathrm{vcd}(\mathcal{N}_g)=2g-5$. Similar results are obtained for the
mapping class group of a non-orientable surface with boundaries and possibly
punctures, and for the pure mapping class group of a non-orientable surface
with punctures and without boundaries.
