Ice boxes - (where a large piece of melting ice was held in a lead tray above the items to be kept cool. The lead tray often had a drain built in to stop meltwater dripping on the items below) were used, but only by wealthy households until cheap ice became available in the Victorian era. Before this ice was collected in the winter from ponds and stored in ice houses. Surprisingly this lasted until mid summer.
Before this storage was normally in larders, which had thick stone walls and were normally below ground and so were always cool....a bit like a cave. But in reality people did not preserve food by cooling it, unless they were well off. Food was preserved by salting it, smoking it (bacon for example, and kippers!), drying it, pickling it, bottling it (preserving fruit by putting it in a jar with water or wine or syrup, then heating it to boiling and putting the lid on. an early version of canning), jam making or preserving in brandy.