In pre-industrial Britain hay was stored in the field in haystacks (of 'needle in a' fame) These were not simply piles of hay but carefully constructed stacks with a thatched roof to keep the content dry and often built on platforms to keep them off the ground. Barns were used exclusively for the storage of corn (corn is a term originally meaning any grain though modern use restricts it to maize, which is not much grown in britain). The american barn appears to be a general store for lots of farm produce, equipment and livestock whereas the british barn was a careful designed building where harvested corn was stored on one side of the building, between two large opposing doors was a threshing floor where the grain was threshed from the corn (the breeze between the doors being used to winnow the chaff away) and the straw then being stored on the other side of the barn. Cattle were kept over winter in a cowhouse, sometimes called a byre, often attached or close to the straw side of the barn. It's worth noting that hay is a separate crop and is an animal feedstuff, whereas straw is a byproduct of grain cultivation and is used for animal bedding.
About as interesting as duct tape then.