Not sure you would classify bank deposits as their "property" - any money you deposit with a bank becomes essentially a loan to them from you which you can ask them to repay depending on how you deposited (lent) it to them either at call or at the end of a term deposit.
The issue of course is the liquidity of the bank in the event of a run where people decide they want their cash out.
Banks by their nature take deposits in and lend out usually with a mismatch in maturities so they never have enough cash available to repay their at call deosits if everything wanted all of a sudden to take their money out - Australian banks do have emergency reserve bank support for liquidity
Anyway if the banking system were to collapse then effectively society would be in a mess as there would be no viable means for any transactions for goods and services which is why in modern times Governments have always supported any banks that got into trouble.
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