Replace that public record with a physical ledger, replace digital signatures with handwritten signatures, and replace proof of work with a requirement to write

Replace that public record with a physical ledger, replace digital signatures with handwritten signatures, and replace proof of work with a requirement to write a haiku of sufficient wittiness and relevance to the lines on the page including the prior page's haiku, and you have a version of Bitcoin that doesn't even need computers. You need something like a telephone party line, too, but that part can be skipped over; it still works if people sit close enough that they form a peer-to-peer earshot network.