The Fairfax, VA library operates like that. You check out the book (which is in ePub) and it's yours for 21 days at the end of which it disappears off your reader. I've read a half dozen or so of their books and it works fine.
For more details on eBook lending libraries, check out our Wiki on the subject here
https://wiki.mobileread.com/wiki/EBook_Lending_Libraries