The EDGES Collaboration has recently reported the detection of a stronger-than-expected absorption feature in the global 21-cm spectrum, centered at a frequency corresponding to a redshift of . This observation has been interpreted as evidence that the gas was cooled during this era as a result of scattering with dark matter. In this Letter, we explore this possibility, applying constraints from the cosmic microwave background, light element abundances, Supernova 1987A, and a variety of laboratory experiments. After taking these constraints into account, we find that the vast majority of the parameter space capable of generating the observed 21-cm signal is ruled out. The only viable models are those in which a small fraction, , of the dark matter consists of particles with a mass of and which couple to the photon through a small electric charge, roughly as large as the electron charge. Furthermore, in order to avoid being overproduced in the early Universe, such models must be supplemented with an additional depletion mechanism, such as annihilations through a gauge boson or annihilations to a pair of rapidly decaying hidden sector scalars.