We combine electromagnetic (EM) and gravitational-wave (GW) information on the binary neutron star (NS) merger GW170817 in order to constrain the radii and maximum mass of NSs. GW170817 was followed by a range of EM counterparts, including a weak gamma-ray burst (GRB), kilonova (KN) emission from the radioactive decay of the merger ejecta, and X-ray/radio emission consistent with being the synchrotron afterglow of a more powerful off-axis jet. The type of compact remnant produced in the immediate merger aftermath, and its predicted EM signal, depend sensitively on the high-density NS equation of state (EOS). For a soft EOS that supports a low , the merger undergoes a prompt collapse accompanied by a small quantity of shock-heated or disk-wind ejecta, inconsistent with the large quantity of lanthanide-free ejecta inferred from the KN. On the other hand, if is sufficiently large, then the merger product is a rapidly rotating supramassive NS (SMNS), which must spin down before collapsing into a black hole. A fraction of the enormous rotational energy necessarily released by the SMNS during this process is transferred to the ejecta, either into the GRB jet (energy ) or the KN ejecta (energy ), also inconsistent with observations. By combining the total binary mass of GW170817 inferred from the GW signal with conservative upper limits on and from EM observations, we constrain the likelihood probability of a wide range of previously allowed EOSs. These two constraints delineate an allowed region of the parameter space, which, once marginalized over NS radius, places an upper limit of (90%), which is tighter or arguably less model-dependent than other current constraints.