Where is it? Cloth Fair, just south of Smithfield Market.
What's quirky about it? It's said to be the oldest surviving house in London, a miraculous survivor of the Great Fire, the Luftwaffe, and the even worse Developers. It's been sympathetically restored quite a bit, apparently, but would be instantly recognisable to its original inhabitants of the 1590s.
Why bike there? Cloth Fair (right) is a pioneering bike contraflow street, so you can legally ride the wrong way up a one way street. The ancient church of St Bartholomew opposite does some funny bun-throwing stuff every Good Friday.