Do old preprints actually exist? It is quite common (in some areas) to see the reference to the paper, which is cited as "to appear" or "in prepapation" but never actually appeared. Such citations are treated in the same way as references to unpublished results, conference talks, etc.

However, this approach doesn`t really work for references to preprints. The results are published. The problem is that old preprints (say, from 1985) are now inaccessible, they are often not digitized.

So, it is not clear how to treat such old preprints and papers based on results from them: should I believe the authors citing it or consider it as non-existent?
You should try to track down the preprint.  Having to track it down is annoying, but it's usually not hard to do by asking around.  If your friends don't have a copy, and the author can no longer be contacted, then you should feel free to write to someone who has cited the preprint to ask if they know where to find a copy.  (You can also try the author's collaborators, thesis advisor, students, etc.)  When I've done this, it has generally led pretty quickly to a copy, and I've always found one eventually, even for unpublished work from 40+ years ago.  If necessary, you can also do things like post online requesting a copy from anyone who has one, but this is typically not necessary.

If you can't find a copy, I would suggest giving the citations you have seen the benefit of the doubt and mentioning the preprint in your paper; it's possible that the citations are mistaken, but not likely.  If you can't verify the claims, you can note that you have not been able to track down a copy, but it is cited in X, Y, and Z.  (You should only say this if you have seriously tried to locate one but failed.)

For some purposes, seeing the preprint doesn't really matter.  For example, you may be citing it merely to assign credit, in a situation where all the factual content can be obtained from elsewhere in the literature.  In that case, it's perfectly reasonable to follow the consensus in the field about credit, without worrying about investigating it yourself.

On the other hand, if your work depends on the actual content of the preprint in a way that cannot be verified without a copy, then you really need to track one down.  Some people may be lazy about this, but that's not a responsible scholarly approach.

The most frustrating situation is when you're told that something you've discovered may be previously known, but you are unable to access the paper that might contain it.  In this case you need to try especially hard to find a copy, to counteract your natural incentive to give up and decide the paper doesn't exist or isn't relevant.