We have investigated the I-V characteristics of strongly disordered ultrathin films of Bi and Sn produced by quench condensation. Our results show that both these systems can be visualized as strongly disordered arrays of Josephson junctions. The experimentally observed I-V characteristics of these films are hysteretic when the injected current is ramped from zero to critical current and back. These are remarkably similar to the hysteretic I-V of an underdamped single junction. We show by computer simulations that hysteresis can persist in a very strongly disordered array. It is also possible to estimate the individual junction parameters (R, C, and Ic) from the experimental I-V of the film using this model. The films studied are in a regime where the Josephson-coupling energy is larger than the charging energy. We find that a simple relation Ic(T) = Ic(0)[1–(T/Tc)4] describes the temperature dependence of the critical current quite accurately for films with sheet resistance ~500 omega or lower. We also find evidence of a vortex-unbinding transition in the I-V taken at temperatures slightly below the mean-field Tc.