'Yard of clay', often hyphenated ('yard-of-clay'), was an informal way of refering to a long smoking pipe in the 17th century in England. Long pipes are more commonly called a churchwarden pipe, although many churchwarden pipes are not made of clay.
Despite the name, these pipes were not nearly a yard long, generally being about a foot in length. The length allows for the smoke to cool before it is inhaled, and may keep smoke and pipe bowl from impeding one's vision, although the later was probably not a consideration in the use of yard-of-clays.
It was a tradition in part of England for a tavern to provide pipes for their patrons, just as the establishment would provide cups and plates. Clay pipes were much cheaper than wooden pipes, and could be sanitized by putting them in the oven -- although this was probably not done often. Thus, yard-of-clay had the connotation of a cheap, shared pipe, a friendly item redolent of good times in smoke-filled beer-halls, but certainly not highbrow.
The phrase yard-of-clay probably comes from an allusion to the yard of tin, the common name for a hunting or coach horn. Confusingly, a yard-of-clay is sometimes called a yard-of-tripe (although this phrase is also used to mean a long face, an ugly face, or a long cut of tripe); this is presumably a bit of rhyming slang.