There were lots of economies that worked this way. One recent example was the Irish Bankers Strike. Most of the banks in Ireland closed because they wanted concessions. The banks gave up on their strike because the overall economy wasn’t affected much because people just paid with what cash they had, and if they needed credit they’d go visit their local pub where the owner would vouch for them.

There’s more examples in David Graeber’s book “Debt: The First 5,000 Years.” Early economies didn’t have money, but they still made it work.