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In spectroscopic studies of unstable nuclei, hydrogen targets are of key importance. The CHyMENE Project aims to provide to the nuclear physics community a thin and pure solid windowless hydrogen or deuterium target. CHyMENE project must respond to this request for the production of solid Hydrogen. The solid hydrogen target is produced in a continuous flow (1 cm/s) by an extrusion technique (developed with the PELIN laboratory) in a vacuum chamber. The shape of the target is determined by the design of the nozzle at the extrusion process. For the purpose, the choice is a rectangular shape with a width of 10 mm and a thickness in the range of 30-50 microns necessary for the physics objectives. The cryostat is equipped with a GM Cryocooler with sufficient power for the solidification of the hydrogen in the lower portion of the extruder. In the higher part of the cryostat, the hydrogen gas is first liquefied and partially solidified. It is then compressed at 100 bars in the cooled extruder before expulsion of the film through the nozzle at the center of the reaction vacuum chamber. After the previous step, the solid hydrogen ribbon falls by gravity into a dedicated chamber where it sublimes and the gas is pumped and evacuated in a exhaust line. This paper deals with the design of the cryostat with its equipment, with the sizing of the thermal bridge (Aluminum and copper), with the results regarding the contact resistance as well as with the vacuum computations of the reaction and recovery hydrogen gas chambers. [ABSTRACT FROM AUTHOR]