Summary: We analyse a Eulerian finite element method, combining a Eulerian time-stepping scheme applied to the time-dependent Stokes equations with the CutFEM approach using inf-sup stable Taylor-Hood elements for the spatial discretization. This is based on the method introduced by C. Lehrenfeld and M. Olshanskii [ESAIM, Math. Model. Numer. Anal. 53, No. 2, 585–614 (2019; Zbl 1422.65223)] in the context of a scalar convection-diffusion problems on moving domains, and extended to the nonstationary Stokes problem on moving domains by E. Burman et al. [Numer. Math. 150, No. 2, 423–478 (2022; Zbl 07493698)] using stabilized equal-order elements. The analysis includes the geometrical error made by integrating over approximated level set domains in the discrete CutFEM setting. The method is implemented and the theoretical results are illustrated using numerical examples.