a flexible pressure vessel is constructed from at least one pair of upper and mating lower dome shaped cell portions . the dome portions are molded from sheets of resilient material and joined together by radio frequency welding or high - strength adhesives . upper and lower passageway portions extend outwardly from each cell portion to the surrounding sheet material . when the cell portions are joined the passageway portions are joined to form a passageway for connection to a valve or another cell . upper and lower rings surround the upper and lower cell portions to provide reinforcement for the cells . first and second blankets of heavy - duty fiber reinforced material are attached over the upper and lower cell portions and stitched in place with heavy - duty stitching extending through the resilient material surrounding the cell portions . cell shaped sponges impregnated with absorbent materials are encased in liquid and gas impermeable plastic tubing and inserted into the cells prior to joining of the cell portions . heat - reflecting plastic film or metal foil is inserted between blankets and the cell portions . the heavy duty stitching is high - pressure loop and lock braiding . the passageway has a cross - section of between 0 . 050 and 0 . 100 inches . an apparatus and method are described for constructing the flexible pressure vessel .