Cut strand of hair longevity?
In order to keep the hair from decomposing, my first assumption is you'd need it in an environment that inhibits decomposers, e.g. dry. We can see in the mummification process that hair may be preserved for hundreds of years. Hair is also composed largely of keratin, however, and in my own search of the literature: (1) we need microbial (bacteria, fungi) keratinases to decompose the keratin in hair [source], and (2) this process is slow. There's a really old article here, from the abstract:


  Keratin, prepared from horn meal, was added to moist field and garden soil and allowed to decompose in the laboratory. The keratin was found to undergo a decomposition resulting in a slow, but steady accumulation of ammonia and nitrate. 35–40 per cent, of its nitrogen was transformed into nitrate after 120 days.


More recent numbers to shed light on a rate of degradation for keratin are either obscure, or untested. This is more so for hair specifically as I couldn't find much more.