The Keller-Segel system is a well-established model to describe chemotactic evolution of some density. A characteristic trait of this model is that the solution may blow up in finite time. From analysis, we obtain only short-time existence of weak solutions, without explicitly knowing the time interval of existence. A posteriori error estimators can be used to infer existence of a weak solution based on numerical simulations and to verify existence of weak solutions on explicitly given time intervals. An a posteriori error estimator is obtained by combining stability results for the Keller-Segel system with a posteriori residual estimates. As a numerical approximation, we choose a finite volume method for the first equation of the Keller-Segel system and a P1 finite element scheme for its second equation.