We propose a generic numerical measure of the inconsistency of a database with respect to a set of integrity constraints. It is based on an abstract repair semantics. In particular, an inconsistency measure associated to cardinality-repairs is investigated in detail. More specifically, it is shown that it can be computed via answer-set programs, but sometimes its computation can be intractable in data complexity. However, polynomial-time deterministic and randomized approximations are exhibited. The behavior of this measure under small updates is analyzed, obtaining fixed-parameter tractability results. Furthermore, alternative inconsistency measures are proposed and discussed.