This does seem odd - the link you gave says "Elib has been decommissioned as a separate package since its useful functions have long since been included in Emacs.", but Elib is the first Google result for "emacs doubly linked lists", and I couldn't find anything about them being in Emacs.
I guess the key word there is "useful", which is a bit subjective. But in all my novice lisp programming I've never needed doubly linked lists, and I've never seen any code that uses them.
So I'd say, just use Elib if you need a doubly linked list, and Elib works - it looks like a well documented library. The code is available here - https://github.com/burakbayramli/kod/tree/master/site-lisp/elib. It's interesting to see such an old library - copyright 1991-1995.