A dry gas seal structure comprising: a housing that houses a fluid in a supercritical state; a rotary shaft that is provided so as to penetrate the housing and is rotary-driven; rotation rings that are provided around the rotary shaft and rotate integrally with the rotary shaft; stationary rings that are provided in the housing, the stationary rings contacting with the rotation rings throughout the circumferences 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 in which one end thereof is connected to a first channel of the housing and an other end thereof is connected to a second channel of the housing and which supplies a portion of the fluid housed in the housing to inlet portions of the sealing gaps and circulates the portion of the fluid in the housing; 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, characterised in that the control unit is configured to determine whether the fluid leaks from the circulating pipe path and is configured to announce to the user of a fact of the fluid leaking in a case in which the occurrence of the fluid leaking is determined, and the control unit is configured to determine whether the fluid leak amount from the sealing gap is within an allowed range and is configured to announce to the user of a fact that the seal leak amount from the sealing gap is not within the allowed range in a case in which the occurrence of the fluid leaking from the circulating pipe path is not determined. a housing that houses a fluid in a supercritical state; a rotary shaft that is provided so as to penetrate the housing and is rotary-driven; rotation rings that are provided around the rotary shaft and rotate integrally with the rotary shaft; stationary rings that are provided in the housing, the stationary rings contacting with the rotation rings throughout the circumferences 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 in which one end thereof is connected to a first channel of the housing and an other end thereof is connected to a second channel of the housing and which supplies a portion of the fluid housed in the housing to inlet portions of the sealing gaps and circulates the portion of the fluid in the housing; 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, characterised in that the control unit is configured to determine whether the fluid leaks from the circulating pipe path and is configured to announce to the user of a fact of the fluid leaking in a case in which the occurrence of the fluid leaking is determined, and the control unit is configured to determine whether the fluid leak amount from the sealing gap is within an allowed range and is configured to announce to the user of a fact that the seal leak amount from the sealing gap is not within the allowed range in a case in which the occurrence of the fluid leaking from the circulating pipe path is not determined.