Recently the EDGES collaboration reported an anomalous absorption signal in the sky-averaged 21-cm spectrum around z=17. Such a signal may be understood as an indication for an unexpected cooling of the hydrogen gas during or prior to the so-called Cosmic Dawn era. Here we explore the possibility that sub GeV dark matter cooled the gas through velocity-dependent, Rutherford-like interactions. We argue that such interactions require a light mediator that is highly constrained by 5th force experiments and limits from stellar cooling. Consequently, only a hidden or the visible photon can in principle mediate such a force. Neutral hydrogen thus plays a subleading role and the cooling occurs via the residual free electrons and protons. We find that these two scenarios are strongly constrained by the predicted dark matter self-interactions and by limits on millicharged dark matter, respectively. We conclude that the 21-cm absorption line is unlikely to be the result of gas cooling via the scattering with a dominant component of the dark matter. An order 1% subcomponent of millicharged dark matter remains a viable explanation.