A continuous storage apparatus holds carriers each having many vertical shelves to be continuously loaded and unloaded by pushers at adjacent positions. The conveyor order of arrival of items is maintained on departure. Shelves hold items to be frozen or cooked during storage by a forward or backward (countercurrent) flow of recirculating cold or hot air within the tunnels of the machine. Carrier motion is provided by lifting and lowering conveyor chains at both ends of the tunnels and by shunting the carriers, supported by wheels or skids on rails along the tunnels, when pushed by an actuator.