A hazardous material storage bank system comprising: a first wellbore extending into the Earth and comprising an entry at least proximate a terranean surface, the first wellbore comprising a vertical portion, a transition portion, and a horizontal portion; a storage area coupled to the horizontal portion of the first wellbore, the storage area within or below a rock formation, the storage area vertically isolated, by the rock formation, from a subterranean zone that comprises mobile water; a storage container positioned in the storage area, the storage container sized to fit from the first wellbore entry through the vertical portion, the transition portion, and the horizontal portion of the first wellbore, and into the storage area, the storage container comprising an inner cavity sized to enclose nuclear waste; a seal positioned in the first wellbore, the seal isolating the storage portion of the first wellbore from the entry of the first wellbore; a monitoring system comprising: a plurality of sensors positioned proximate the storage container in the subterranean formation, at least one sensor of the plurality of sensors comprising a radioactivity sensor, wherein the plurality of sensors are positioned within a second wellbore (140, 142, 144) that is formed separately from the first wellbore, and a monitoring control system communicably coupled to the plurality of sensors to receive radioactivity data from the at least one sensor. a first wellbore extending into the Earth and comprising an entry at least proximate a terranean surface, the first wellbore comprising a vertical portion, a transition portion, and a horizontal portion; a storage area coupled to the horizontal portion of the first wellbore, the storage area within or below a rock formation, the storage area vertically isolated, by the rock formation, from a subterranean zone that comprises mobile water; a storage container positioned in the storage area, the storage container sized to fit from the first wellbore entry through the vertical portion, the transition portion, and the horizontal portion of the first wellbore, and into the storage area, the storage container comprising an inner cavity sized to enclose nuclear waste; a seal positioned in the first wellbore, the seal isolating the storage portion of the first wellbore from the entry of the first wellbore; a monitoring system comprising: a plurality of sensors positioned proximate the storage container in the subterranean formation, at least one sensor of the plurality of sensors comprising a radioactivity sensor, wherein the plurality of sensors are positioned within a second wellbore (140, 142, 144) that is formed separately from the first wellbore, and a monitoring control system communicably coupled to the plurality of sensors to receive radioactivity data from the at least one sensor. a plurality of sensors positioned proximate the storage container in the subterranean formation, at least one sensor of the plurality of sensors comprising a radioactivity sensor, wherein the plurality of sensors are positioned within a second wellbore (140, 142, 144) that is formed separately from the first wellbore, and a monitoring control system communicably coupled to the plurality of sensors to receive radioactivity data from the at least one sensor.