As so often happens, it turns out that older members in our society knew something we didn’t. We now know that a fly net was worn by a horse. Obviously, it could not catch the flies, but its movement with the horse's would help to keep flies from settling. According to W. R. Runyan’s 1001 Words and Phrases You Never Knew You Didn’t Know (2011), the large openings were necessary to keep the horse from overheating.
Spread out, the fly net turned out to have leather pieces attached to the strings, long strings hanging from each side, and braids at one end.
And now we know not to dismiss a heap of string in our barn.