  We construct and analyze a CutFEM discretization for the Stokes problem based
on the Scott-Vogelius pair. The discrete piecewise polynomial spaces are
defined on macro-element triangulations which are not fitted to the smooth
physical domain. Boundary conditions are imposed via penalization through the
help of a Nitsche-type discretization, whereas stability with respect to small
and anisotropic cuts of the bulk elements is ensured by adding local ghost
penalty stabilization terms. We show stability of the scheme as well as a
divergence--free property of the discrete velocity outside an $O(h)$
neighborhood of the boundary. To mitigate the error caused by the violation of
the divergence-free condition, we introduce local grad-div stabilization. The
error analysis shows that the grad-div parameter can scale like $O(h^{-1})$,
allowing a rather heavy penalty for the violation of mass conservation, while
still ensuring optimal order error estimates.
