An important feature of many real world facility location problems are capacity limits on the number of agents served by each facility. We provide a comprehensive picture of strategy proof mechanisms for facility location problems with capacity constraints that are anonymous and Pareto optimal. First, we prove a strong characterization theorem. For locating two identical facilities with capacity limits and no spare capacity, the INNERPOINT mechanism is the unique strategy proof mechanism that is both anonymous and Pareto optimal. Second, when there is spare capacity, we identify a more general class of strategy proof mechanisms that interpolates smoothly between INNERPOINT and ENDPOINT which are anonymous and Pareto optimal. Third, with two facilities of different capacities, we prove a strong impossibility theorem that no mechanism can be both anonymous and Pareto optimal except when the capacities differ by just a single agent. Fourth, with three or more facilities we prove a second impossibility theorem that no mechanism can be both anonymous and Pareto optimal even when facilities have equal capacity. Our characterization and impossibility results are all minimal as multiple mechanisms exist if we drop one property.