In this paper, we present cut finite element methods for the Stokes equations. They are designed to satisfy pressure-robustness, which in the finite element context means having velocity error estimates independent of the pressure error. In the fitted finite element framework, where boundary conditions are enforced strongly, using finite element velocity-pressure pairs that yield pointwise divergence-free velocity fields is sufficient to generate pressure-robust schemes \cite{john_divergence_2017}; however, this is not the case in unfitted finite element methods, where the boundary conditions are imposed weakly. In this work, we establish conditions under which pressure-robustness can be achieved in cut finite element methods. This is illustrated by several numerical examples, both for the fictitious domain and interface Stokes problem, and is motivated by theoretical analysis.