Couple of dozen, give or take. Depends on how you define "hostel." There's no formal designation. Some have been around forever but others come and go. FreeState hostel in MD was wonderful -- but that was ten years ago, and they formally closed down a year or two back. Frats and dorm rooms at Dartmouth? No more, I think. Terrapin Station hostel near Front Royal has been on and off for a couple years now. The list changes every year.
Some are free but offer no services, other than a roof over your head. Jim Murray's place, and Dan Quinn's.
Elmers', NOC, Woods Hole, Ms. Janet's, Kincorra, Teahorse, The Doyle, Hikers Welcome, Shaws... these are perennial, almost. Just a small sampling. But any one of them could shut down "without prior notice" as they say.