Edit: btw, we really need a public and free payment system, why are we giving to paypal and credit cards this incredible privilege? Paying with digital cash should be as inexpensive as paying with real cash.
Think about time spent counting, ensuring that there are no falsified notes, bringing excess notes to the bank and making sure there is change, etc.
I think it is quite feasible to create an electronic payment system that has comparable overhead.
Consider that there are existing electronic payment systems, e.g. electronic cash in europe, that already work and are quite secure (being smartcard-based)
The only way besides having centralized for-profit organizations maintaining the payments infrastructure is to use cryptocurrencies.