I believe Birmingham Central library still has their records prior to 1972(?) on just such a system since they've never been migrated to the computer system which contains everything after that date.

But that's beaten hands down by the Birmingham & Midland Institute, which was the city's original public library in the 19th century. All their books are on index cards. Not only that but members have one or two bigger cards themselves (one for books and one for CD/LPs) on which your borrowings are written by hand. If you ever want to renew an item, they have to write it in as returned then write it out again as borrowed on your card. There's, in theory, a month loan period, but since there are no date stamps in the books/CDs, I never now how long I've had anything. Until I get a phone call from someone that is - I got one before Christmas saying that I'd borrowed a book in June so I'd had it rather a long time, could I return it? I did so a couple of weeks ago. There aren't any fines so the most they can do is make you feel guilty about depriving other members. 

The BMI have had to computerise their CDs however, since the discs are shelved according to their reference numbers and these consist of a prefix (which refers to the source/donor) plus a sequence number. Without the printed list which they've got, it would take ages to find any particular CD, since it could be anywhere.