Some scuba divers hand feed them which I think is fairly stupid and asking for trouble. It's a bit like they hand feed stingrays in some foreign dive centers. How good an idea is that? Ask Steve Irwin.
I believe that the stingrays (if in captivity) have there barbs removed. there are little (and big) morays all over the place where I dive and I have had some small ones approach me while I gut my fish in the shallows it's good fun to feed them some scraps or guts. I've fed some big ones but never with my hands.
As far as congers go I encounter what I assume to be the same one almost every time I night dive at my home spot. he is so calm he'll let me pet him if I approach it slowly enough. You can do the same thing with porcupine fish it's very cool. I don't recommend either btw.