The Erdenet mine (Erdenet, Mongolia) is a copper–molybdenum open pit mine with a huge tailing pond and is located next to a residential area. As the sources of drinking water in this area rely on groundwater and can be categorized into public or private wells, we aimed to assess the groundwater quality of the different types of drinking water wells. To accomplish our goal, we analyzed 18 trace metals (As, Ba, Bi, Cd, Co, Cr, Cs, Cu, Ga, Mo, Ni, Pb, Se, Sr, Th, U, V, Zn) and 8 major metals (Al, Ca, Fe, K, Mg, Mn, Na, Si) samples using ICP-MS and ICP-OES, and the heavy metal pollution index, heavy metal evaluation index, and degree of contamination were calculated. The most concerning point is that the Mo concentration of waste water in the Erdenet mine tailing pond and at a natural spring located 300 m from the tailing pond was 1100 μg/L, which greatly exceeds the WHO permissible concentration for Mo of 70 μg/L. Furthermore, high Ca and Mg concentrations at all private wells indicated that consumers of these wells are more vulnerable to any type of water pollution, as these wells are not built with any disinfection or treatment system. A modified categorization of the water quality indices showed that the public and private wells have low–medium contamination and that the tailing pond seepage water, along with its nearest spring, have a high degree of contamination. Because many private wells are located along the tailing pond and used directly without any treatment, we suggest a continuous evaluation and monitoring of the groundwater quality in this area.