The invention relates to a reinforced concrete stair structure and a construction method of the reinforced concrete stair structure. The construction method of the reinforced concrete stair structure comprises the steps that stair boards, a stair beam and formworks of platforms are erected; stair board rebars, stair beam rebars and rebars of the platforms are bound, and upper rebars of the stair board rebars, the stair beam rebars and the rebars of the platforms are bound together; concrete pouring is carried out, wherein stair beam concrete is poured to form the stair beam, platform concrete is poured to form the platforms, and the stair board concrete is poured and trowelled to form step structures; buffer structures are arranged between the bottoms of the stair boards and installation surfaces respectively, wherein each buffer structure comprises a movement gap and a first isolation layer arranged on the corresponding installation surface; after the poured concrete solidifies, the stair boards, the stair beam and the formworks of the platforms are disassembled, so that the reinforced concrete stair structure is formed. The construction method of the reinforced concrete stair structure is simple, high in operability and low in cost, and structural damage caused by earthquakes can be avoided.