"How can broke economies lend money to other broke economies who haven't got any money, because they can't pay back the money the broke economy lent to the other broke economy and shouldn't have lent in the first place, because the broke economy can't pay it back?"First of all, this is not necessarily a new phenomena. Many times in the past, countries have borrowed money they couldn't repay. The old solution to this problem was war: go to the country that owes the money and conquer them, taking their land, crops, gold, etc... Fortunately, I don't think German troops are heading to Greece to claim the Parthenon any time soon.
So what will happen? Where does money come from? The answer is that the European Central Bank, along with other central banks, will "print the money". This is what the US Federal Reserve did in response to the financial crisis in 2008. In our current global economy, there is no intrinsic value to money. It is created by governments to facilitate trade in goods and services.
I am not saying that all of this money creation is a good thing, just that it is likely to happen. Right now, there is a struggle in Europe between the Germans and everyone else around this issue of money creation. One of the most likely effects of this would be inflation, which is something the Germans have an institutional fear of (Think World War I and the Weimar Republic).
So this all goes back to the discussion I've had over the past several months about government debt in general. One way out of a debt problem is to cause inflation, which may sound counter-intuitive to many. This makes the debt owed worth less, because debts incurred today will be paid back with currency that has lost purchasing power due to inflation. For example, if I owed you $10 but paid you back your $10 five years from now when it would only by $8 worth of goods and services, then I've "saved" $2. This is not a free lunch, and the price of inflation is paid by those who have savings and/or live on pensions or fixed incomes. This means that it is imperative for retired people to keep some of their assets in investments that will earn some positive return above inflation, such as equities have done historically. Socking away all of one's assets in bonds and cash can be disastrous when we are facing the potential for significant inflation in our not so far off future.
Of course, central banks could fail at their attempts to inflate. The economy is a very complex, unpredictable thing, and the digital revolution, along with China's re-awakening, have made things even more complex and difficult to predict.