What would happen if every person in the world pulled their money out of their bank accounts?
In principle, not that much would necessarily happen: if people pulled out their bank deposits in the form of cash, then the central bank would just become a much bigger financial intermediary.
Cash is a liability of the central bank: when it introduces cash into the economy, it uses that cash to buy an asset. In a world where the public switched to cash en masse, a central bank fulfilling its usual operating procedure by targeting short-term interest rates would fully accommodate that spike in demand for cash by making much more of it, and accumulating assets in the process. What would these assets be? Well, in this situation they would naturally be loans to banks, which would need the funding.
So by withdrawing its funds from the banking system, the public would just trade off holding deposits directly in banks for holding these assets indirectly through the central bank (by holding cash that is backed by loans to those banks).
Of course, this is all a bit idealized. In practice a mass withdrawal of funds would presumably be somewhat chaotic, and the central bank would have to be diligent about its role as lender of last resort to avoid disaster. But there is a natural way for the system to work here, and it's amusing to contemplate how in this scenario, we'd really just be adding one more layer of indirection to the financial system.
The thing with these economics questions that never happened empirically is always the same: nobody really knows. Befor Creditanstalt 1931 nobody thought that a bank can go bancrupt, during BrettonWoods/Goldstandard freely floating exchange rates were sought of being detrimental and dumb....
But in theory everything should collapse due to our fractional reserve system that requires banks to have a certain amount of money stored in their books for every granted loan. But thats not going to happen because centralbanks would intervene and convince people that they are going to do anything to keep the banking system working. If the CB is able to make a trustworthy commitment that should work. Because a rational individual should have no incentive to have so much cash... The costs of carrying so much money are simply too high...
Once there was this Eric Canton attempt....
A such systemic bank run would be caused by a massive loss of trust in the banking system and financial markets. Its a collapse and bankrupt of banks possibly capitalism system as we know it. It would be a new era.
New institutions would emerge. Possibly decentralized ones like Bitcoin. If there is no possibility of international payments, international trade would be almost non existent.
How would credit would exist in such a world without money creation and central banks? Majors big investissements would be difficult.
In fact I don't think people would withdraw money in the first place. In a total loss of confidence money would be worth nothing. Perhaps metals would be back in fashion for trades. Perhaps a cooperative system of reputation like Stackoverflow...
There exist a realistic possibility of collapse. What if a virus enables automatic payment system like interact or cirrus? In an instant destroy the banking system before people could actually withdraw.
In any case I do not forsee a calm transition to a new system. Forget rational behaviors then.