Short baseline neutrino oscillation experiments have shown hints of the existence of additional sterile neutrinos in the eV mass range. Such sterile neutrinos are incompatible with cosmology because they suppress structure formation unless they can be prevented from thermalizing in the early Universe or removed by subsequent decay or annihilation. Here, we present a novel scenario in which both sterile neutrinos and dark matter are coupled to a new, light pseudoscalar. This can prevent thermalization of sterile neutrinos and make dark matter sufficiently self-interacting to have an impact on galactic dynamics and possibly resolve some of the known problems with the standard cold dark matter scenario. Even more importantly it leads to a strongly self-interacting plasma of sterile neutrinos and pseudoscalars at late times and provides an excellent fit to cosmic microwave background data. The usual cosmological neutrino mass problem is avoided by sterile neutrino annihilation to pseudoscalars. The preferred value of H-0 is substantially higher than in standard Lambda CDM (lambda cold dark matter) and in much better agreement with local measurements.