The exciton spin dynamics are investigated both experimentally and theoretically in two-monolayer-thick GaAs/AlAs quantum wells with an indirect band gap and a type-II band alignment. The magnetic field induced circular polarization of photoluminescence Pc is studied as function of the magnetic field strength and direction as well as sample temperature. The observed nonmonotonic behavior of these functions is provided by the interplay of bright and dark exciton states contributing to the emission. To interpret the experiment, we have developed a kinetic master equation model which accounts for the dynamics of the spin states in this exciton quartet, radiative and nonradiative recombination processes, and redistribution of excitons between these states as result of spin relaxation. The model offers quantitative agreement with experiment and allows us to evaluate, for the studied structure, the heavy-hole g factor, ghh=+3.5, and the spin relaxation times of electron, τse=33μs, and hole, τsh=3μs, bound in the exciton.