2nd the whole Hoxton/Shoreditch/White Chappel/Brick Lane suggestion (all very georgraphically very close, at the intersection of the london borroughs of Hackney and Tower Hamlets).
Bits of Tower Hamlets and Hackney are still affordable. Some nice concentrations of shops/people around Newington Green, London Fields and Brick Lane areas but prices just keep going up. If you like it a little edgier then Hackney Central and Dalston have lots to offer.
Main reason Hackney's cheap is that theres no tube in the borrough. But there will be soon (although its just an extension of the joke that is the East London Line). But it has pretty good Rail and excellent bus connections, as well as being a short 15 min ride away from central London.