The fabric is linen, which comes from flax plants. C-14 dating would essentially be dating when the flax plants were harvested (that is, killed). That’s when they would have stopped absorbing C-14 containing CO2 from the air.
You’ve been most kind in your reply. I still balk at the notion carbon dating can distinguish between C-14 absorbed on the last day of an organisms life and C-14 present at the first day of creation. For example, the water absorbed in my body, should I die today, is certainly older than my body, no?