Do it right. Use adhesive tape designed to work with fabrics, like some bias tapes. Nothing new under the sun. Or just take whole cloth as it is - the ends won't fry.
You might get lucky if the cloth is loose enough, so that the tape would be able to bind to itself through it.
I'm quite sure that's first duct tape hammock I've seen, I'm also sure that's not a first no-sew option, as those are probably around since people moved into the jungle