What would a country look like without banks? In 1970, all the banks in Ireland closed for half a year. In response, the Irish people set up their own exchange systems centred on (of course) pubs.
At the end of the 1960s, the cost of living went up a lot in Ireland. Most employers increased workers’ wages to account for this increase, but the four major Irish banks (the “Big Four”) did not. So the banking staff went on strike… and it was an enormous strike. All unionised banking staff, in all the banks in Ireland, stopped working at the start of May, 1970. They stayed on strike for more than six months.
1970, of course, is well before digital banking. Cash and cheques were the primary means of buying and selling. And, all of a sudden, no-one could get any money out of (or into) their bank accounts. Houses could not change hands because the property deeds were all locked up. You would expect the whole country to go into a kind of economic and social meltdown. But, against all odds, it did not.
Instead of disaster, the people of Ireland began writing each other promissory notes and custom cheques. And these notes became a kind of currency of their own, moving between people who took the cheques’ value on faith. For cash, pubs became the central clearing house: people would cash their cheques in the pub and then stay to have a drink. Win-win. And because the pub was such a central part of the local community, they could operate on a trust model very effectively.
It was not, of course, a perfect system. Without access to bank records, people could make promises they could not keep:
A certain racehorse trainer was short of money but wanted to buy a horse. He gave the vendor a cheque even though he knew he didn’t have the funds to meet it. Fortunately for him the horse won a number of races so he had no trouble honouring the cheque and making a handsome profit without having to pay a penny interest for a very risky venture.
How six-month bank strike rocked the nation
But on the whole, Ireland carried on surprisingly well without any banks. The strike ended in November, and is remembered now mostly as a case study for just how a country might operate without a banking sector.
[Thanks to Gareth E.]
The Generalist Academy 
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I think this is an important example in Rutger Bregman’s “Utopia for Realists”, where it is explained with some glee how ineffectual this strike was (in contrast to, say, garbage workers’ strikes). He is a man to watch.