  We define the notion of a hierarchically cocompact classifying space for a
family of subgroups of a group. Our main application is to show that the
mapping class group $\mbox{Mod}(S)$ of any connected oriented compact surface
$S$, possibly with punctures and boundary components and with negative Euler
characteristic has a hierarchically cocompact model for the family of virtually
cyclic subgroups of dimension at most $\mbox{vcd} \mbox{Mod}(S)+1$. When the
surface is closed, we prove that this bound is optimal. In particular, this
answers a question of L\"{u}ck for mapping class groups of surfaces.
