Packaging of respiring biological materials, particularly bananas and other fruits, in sealed containers. The containers preferably include a gas-permeable membrane comprising (1) a microporous film, and (2) a polymeric coating on the microporous film. Using appropriate containers and appropriate controlled atmospheres around the container, the respiring materials can be stored and/or ripened under controlled conditions. Bananas can be ripened while they are being transported, or in conventional ripening rooms without opening the containers in which they have been transported. The ripe bananas are less dehydrated and remain in a satisfactory ripened state for longer periods of time.