I've read some non-mainstream theories that economies based on cash accounts and equity-only investments are possible. Some would argue that equity-only investment arrangements are better for society, as they align interests between capital and production in a way that recourse interest lending cannot.

You can already see some fintech startups attempting equity-based home "loans", in which the "lender"'s lien is on a percentage of the future sale value of the home, not a fixed dollar amount. It will be interesting to see how this develops.

I'm not sure how financing on consumer debt like credit cards and non-commercial vehicles would work, though. Perhaps discouraging consumer profligacy would be a feature, however, not necessarily a bug.

In utopian reality, you could run a state owned bank that was run not-for-profit and only charged interest enough to cover losses and expenses.

By barter? By producing something valuable and exchange it through fair trade, and using sound money (not worthless currency)