The dry gas seal structure has a housing that houses a fluid in a supercritical state, a rotary shaft that penetrates the housing, rotation rings that rotate integrally with the rotary shaft, stationary rings that contact with the rotation rings when the rotary shaft is stopped and stationary positioned in a state in which sealing gaps are formed between the rotation rings and the stationary rings when the rotary shaft rotates, a circulating pipe path that supplies a portion of the fluid to inlet portions of the sealing gaps, a temperature-adjusting device that adjusts the temperature of the fluid flowing through the circulating pipe path, and a control unit that controls operation of the temperature-adjusting device so as to heat the inlet portions of the sealing gaps to a predetermined temperature before the rotary shaft is driven.