A fuel cell stack structure and a method of starting up the fuel cell stack structure are disclosed. The fuel cell stack structure includes a stack of a plurality of solid electrolyte fuel cells, each equipped with a solid electrolyte simplex cell accommodated in a cell space surrounded by a metallic thin plate-like separator and having one surface exposed to the outside, and a gas flow channel formed in and extending through the solid electrolyte fuel cells to supply gas to the respective cell space areas of the solid electrolyte fuel cells, wherein an area with high-heat capacity is preferentially supplied with and heated by high temperature gas at the stage of increasing temperatures of the plurality of solid electrolyte fuel cells during startup thereof.