I'm surprised nobody mentioned horse hair, once upon a time fly line, the only line available, was made out of horse hair. Poor horsey.
I don't know about fly lines being made with horse hair, but they were made with silk and treated to enable them to float. I've read about horse hair being used for leaders for flies, as well as "cut gut" and silk, all of which were tied to eyeless hooks.
Silk fly lines are still available, mostly for folks who want the "traditional" to use with bamboo rods, but they're not a mainstream item now and certainly not cheap. But, I guess if a person can afford the bamboo fly rod, they can likely afford such a line.
Frankly some of the modern fly lines today have become crazy in price, with some over $100. Needless to say, I own none in that price range!
Dacron also is very much available and is still used for trolling and bottom fishing. Some fly line backing is dacron, and some made with gel spun polyethylene. The advantage is supposed to be strength per diameter and GSP allows more capacity on a reel. Like many other of to days modern braids it's costly. I'm sure it's a good choice for some folks and the fishing that they do, just like lines used for bass are a good choice.
I had purchased a big 1200 yard spool of dacron line years ago that I use for backing with some of my fly reels. I've read that fly line backing dacron, is slightly less diameter, but frankly have never seen that to be of great importance for the fly fishing I've done. I have some fly reels with Cortland "Micron" dacron backing and the regular dacron does the job just as well, and costs slightly less.