This is the inner lining of the Gemini spacesuit that NASA astronaut Virgil "Gus" Grissom used for training when preparing for his Gemini 3 space mission, his second flight in space. The B. F. Goodrich company made this early prototype under contract to NASA. They made three subsequent iterations of the suit that flew during the Gemini missions. Without the white, betacloth coverlayer, one can see the outer workings of the spacesuit's pressure layer.
NASA transferred this suit to the museum after they had finished the Gemini program.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.