Here's a figure to chew on, though you may find yourself choking. It comes from the US, but there are similar numbers in the UK, too. Remember those banks that were so close to collapse, so desperate and needy they held out the begging bowl and pleaded with the taxpayers for help? Well, it turns out that nine of those banks – who between them trousered $175bn of the American public's money in bailout funds – have fallen back into their old habits: last year they paid their top staff $32.6bn in bonuses.

You read that right: the number is in billions, not millions. Institutions that were so broke the government had to raid the public piggybank to help them have rewarded their employees – the geniuses who drove the world economy off a cliff – with a $32.6bn shower of cash. That includes the truly incredible $98.9m paid to Citigroup's master of the universe, Andrew Hall, along with 5,000 lucky individuals who received more than $1m each.

In the City of London, payday is looking just as golden. Half of Goldman Sachs's first-quarter profit of £1.2bn has already been set aside for staff rewards. No wonder City traders, so briefly forced to bow their heads in shame, are once again puffing out their chests and revelling in the buzzword of the hour: BAB – bonuses are back.

Our politicians are promising to act. The shadow chancellor, George Osborne, says bonuses for banks that have benefited from government help are "unacceptable". Alistair Darling insists he'll move too, though not against all stratospheric bonuses – just those that pose a "systemic risk" to the entire financial sector. Which should let a few fatcats off the hook. And when both Compass and Vince Cable proposed a high pay commission this week, Darling and Osborne ran a mile. In a sentence that warns those who hoped Labour might finally crack down on lunatic pay not to hold their breath, Darling said: "I think that pay agreements ought to be reached by employers and employees meeting together."

It's as if our leaders still see bankers' megabucks as an iron law of nature that cannot be challenged. What are we to do, beyond howling at the moon in our rage? The answer might be to look for different, less conventional remedies. I've written already of the growing campaign for a revival of ancient usury laws, designed to place a legal limit on the interest banks can charge. But here's an idea that might be even more radical: why not do away with banks altogether?

This is no utopian fantasy from a leftist groupuscule, nor an abstract musing from the academic seminar room. It's a business model, one that is already up and running. It responds to this latest crisis of capitalism by wielding capitalism's most lethal weapon: competition.

Enter Zopa, a website that describes itself as a place "where people meet to lend and borrow money … sidestepping the banks". The idea is pretty simple. Someone who has money to spare goes online, says how much he's ready to lend and at what rate of interest – and waits for would-be borrowers to take him up on his offer. If both sides are happy – and Zopa stands for the negotiating term "zone of possible agreement" – then the deal goes ahead. Quite a few of them, as it happens: Zopa has now facilitated £50m worth of loans, from one ordinary Briton to another.

The theory is that everyone benefits, the lender enjoying a much higher rate of return than he would from a regular savings account, and the borrower paying off his debt at a much gentler rate of interest. That's not difficult, says Zopa, when the high street banks are being so stingy towards savers and so demanding of borrowers. Current deals on Zopa are running somewhere between 8% and 10%, while savers would be lucky to earn more than a few points in interest and borrowers can be looking at charges in the teens or higher.

It sounds nice – a woman in Peterborough with a few bob lending to a bloke in Dundee who wants to replace his car – but surely it's fraught with risk. What if our man in Dundee can't or won't pay back the loan? Zopa's answer is that would-be borrowers undergo the same credit checks as imposed by any bank. Besides, the lady in Peterborough is not really lending all her money to that one chap in Dundee: if she's lending £500, Zopa will chop that up into 50 parcels of £10 each, spreading her risk across 50 different people. If one of them refuses to pay up – and so far only 59 out of 10,000 loans have defaulted – she'll be down by £10, not £500. And Zopa promises to chase the defaulting borrower as zealously as would any bank.

That said, Zopa boasts that its greatest strength is that it is not a bank: it's merely holding the ring for people to come together, peer-to-peer as internet folk like to put it (in return for which the company takes a £118.50 fee from the borrower and a 1% annual fee from the lender). "Banks are staggeringly inefficient machines," says Zopa's founder, Giles Andrews. They charge so much because they have enormous overheads, "ancient IT systems, high street branches, directors paying themselves millions". The rules require banks to have reserves of cash, money they "leverage" against when they start playing in the global capital markets casino. But Zopa is not in that game, so doesn't need to maintain that cushion of billions.

There's no logical reason why websites like Zopa – and its exotically named international counterparts Prosper, SmartyPig, Wonga, Mint, Wesabe – couldn't do to banks what iTunes has done to record shops: make them all but irrelevant. Zopa is more cautious, identifying some tasks that will always need banks. The company can't envisage peer-to-peer mortgages, for example: how many individual lenders would want to tie up their money for 25 years? Nor is it easy to imagine the website that could corral enough punters to provide the cash for a heavy-duty infrastructure project, one that requires billions but takes decades to pay back.

But for the smaller scale chunk of banks' business – the personal loan – it is perfectly possible to foresee the day when the big boys are pushed aside, priced out by ordinary people borrowing and lending at rates the banks can't manage. It's a delicious thought: rapacious institutions, currently regarded by governments as too valuable to fail and given permanent guarantees of taxpayer support, going the way of the dodo.

It's especially pleasing that the source of this threat is an idea with a long, progressive lineage. Take away the internet whizz-bangery and Zopa is a latter-day credit union, if not a friendly society, resting on the simple but radical notion of individuals helping each other out. They may not be tied by geography, as they were in the days of Victorian mutuality – and still are in the microfinance schemes that brought Muhammad Yunus and his Grameen bank of Bangladesh the Nobel peace prize. Instead of villagers vouching for the character of one of their neighbours, we have the Equifax credit rating agency. But the principle is the same.

Some leftists might choke on this emphasis on the individual rather than the collective, others on all this talk of profits and interest rates. But they should recognise that there is an enlightened idea at work here. One that might make those bankers a tad uneasy, even as they count their billions.