a flexible pressure vessel is constructed from at least one pair of upper and mating lower dome shaped cell portions . upper and lower passageway portions extend outwardly from each cell portion to surrounding sheet material . the portions are joined to form a passageway for connection to a valve or another cell . upper and lower reinforcing rings surround the cell portions . blankets of fiber reinforced material are attached over the cell portions and stitched in place through the surrounding resilient material . cell shaped sponges impregnated with absorbent materials are encased in impermeable plastic tubing and inserted into the cells . high - strength filaments are wound around the tubing to provide additional pressure handling capability . heat - reflecting plastic film or metal foil is inserted between blankets and the cell portions . reinforcing rings are attached over and fastened through the blankets around the cells . an apparatus and method are described for constructing the flexible pressure vessel .