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.