In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ℙ 2 − ℙ 1 Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L 2 ‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.