I was studying archaeology 10 years ago - and back then, researchers were moving their funds away from C-14 dating and towards other, indirect methods (such as dendrochronology, which dates artefacts using tree-rings), precisely because radiocarbon was becoming increasingly seen as insufficiently accurate. It can give you a ball-park date for an artefact, providing that artefact is not supposed to be more than about 70,000 years old, but with accuracy routinely worse than +/- 250 years (often much worse) it's useless at dating something so precisely fixed in time as the Turin Shroud.
I think the context of the Turin Shroud is far more telling as a clue to its likely accuracy (or otherwise). It belongs to a church movement which collectively owns more vials of milk from the breast of the virgin than could possibly be considered decent (I mean, what was she doing expressing her milk in the first place? Somehow I can't see Joseph getting up to do the 2am feed)
Relics, and their control, was key to the authority of the medieval Roman Catholic church. Seen in that light, the Turin shroud is just too convenient for me.
EDIT: Also seems strange to me that the face on the shroud looks like a 'western' man painted in the medieval stylee, rather than an anatomically accurate semitic man. Compare:
Turin Shroud:
Effigy of Henry III: (13th century)