This is a lean-to shed with an open front. It’s mainly an English West Country word, from the counties of Somerset, Dorset, Devon and Cornwall. However, it has also been recorded in Northern Ireland and Northumberland, which suggests it derives from an ancient English word once more widely distributed.
The Oxford English Dictionary tentatively suggests the Old English hlinian, to lean. A related West Country word, mowhay, means a stack-yard or other enclosure. A mow is a heap of some item, usually a useful crop such as hay, wheat, or barley (it seems to be related to words in Swedish and Norwegian but is otherwise obscure in origin). The second element is from Old English and means an enclosed space (it derives from the same root as hedge, which could at one time mean any enclosing barrier, not necessarily a row of bushes or small trees). So a linhay is a leaning enclosed space.
It has largely been the preserve of writers about the West Country, such as R D Blackmore, in whose Lorna Doone, set on Exmoor, the word may be found a number of times: “That faithful creature, whom I began to admire as if she were my own (which is no little thing for a man to say of another man’s horse), stopped in front of a low black shed, such as we call a ‘linhay.’” Thomas Hardy used it many times in his books set in the fictional Wessex, in reality Dorset. This is in The Return of the Native: “To dissipate in some trifling measure her abiding sense of the murkiness of human life she went to the ‘linhay’ or lean-to-shed, which formed the root-store of their dwelling and abutted on the fuel-house.”
Other writers with West Country connections who used it include Eden Phillpotts, John Galsworthy and Sir Arthur Quiller-Couch. Like Blackmore and Hardy, most felt the need to explain this dialect word. These days it seems to crop up most often in the names of self-catering cottages converted from old barns.
An older spelling is linny or linney, which more accurately reflects the way it is said.