We investigate some consequences if neutrinoless double beta decays (0 ν β β s ) of nuclei are dominated by short range interactions. To illustrate our results, we assume that 0 ν β β s proceed mainly through short range interactions involving two-W-boson exchanges and confine ourselves to only include new scalars without new gauge interactions for the SM fermions. For the neutrino mass problem, we propose to solve it by adopting that the light neutrinos have predominantly Dirac masses and the small Majorana masses induced by the new scalars render them quasi-Dirac particles. This particular aspect of neutrinos may be detectable in the next generations of neutrino oscillation experiments and/or neutrino telescope. If so this opens a new connection between 0 ν β β and neutrino physics. We also note the new physics signals such as the high charged states that can be explored in hadron colliders. In particular, we find that a high energy e-e- will be very useful in testing the origin of lepton number violation, which complements the 0 ν β β studies.