In Israel, we used to have post-office issued debit cards. You buy the card at the counter, with cash, and then it could be used for online payments, until you exhaust your credit. This is not entirely different from a bank, of course, but it allowed users to separate their purchases history from their income history.

Post office is also used for things like utility payments. You can pay them online, or write down the invoice id, the recipient, and pay with cash at the post office. I'm not sure how it works from the recipient perspective: i.e. how would you register your business with the post office, but, the existence of some such payments implies it should be possible.

NB. Also, technically, these services while physically provided in the post office are sometimes referred to as being provide by "postal bank". I'm not sure why it's called a bank though, since it doesn't provide key banking services s.a. loans or check-in accounts... Maybe it's just a "historic" name.