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The Korea Basic Science Institute has been developing a 5-T high temperature superconducting (HTS) insert magnet system. The objective of the program is the design and fabrication of an HTS insert magnet system for high-field superconducting magnet applications. Because the HTS magnet system should be inserted into the conduction-cooled 15 T background magnet with 102-mm room-temperature bore, several ancillary requirements such as space restriction and cryostat structure need technical attention. In this paper, the design of the HTS insert cryostat for 5 T HTS coil is presented. The design process of the novel cryostat to overcome the cryogenic space restriction is described, taking into account the dimension of the HTS coil, cryogenic loads, and inventory of cryogen. The structure of the vessel, considering support structure and thermal path, is also described. In addition, the fabrication of the insert cryostat is briefly presented.