There's someone around here who spins hair from her Pyrenees. She's not the only one, either - you can seriously google "dog hair yarn" and you'll come up with a surprising number of hits.
There's a culture in Asia - I think Mongolians? - who keep large hairy dogs and spin their hair - beautiful dogs and they are treated like members of the family. In fact if food runs low they'll feed the dog first before the kids, so really the dogs are better treated than the family! But the dogs are super important - they keep the horses safe, help warm the houses, and of course there's the hair they use for all kinds of stuff.
I'm actually allergic to all natural fibers, except angora goat and rabbit. Weird since I'm a sheep farmer, but there it is. I still wear cotton and wool, however - I hate polyester (ironically). I think I'd have to draw the line at dog, though.