How can I privately insure against bank going bankrupt?

In the United States you can open accounts at different banks/Credit Unions. In The US the amount of coverage can be increased by the use of Joint and Individual accounts. A couple can get 3x the coverage by each having a personal account, and then having a joint account.

In Australia this doesn't work the same way.

For joint accounts, the amount of FCS protection is determined by splitting the deposit equally between the account holders. Each account holder's share is then added to any other eligible deposits they may hold under their own name at the same bank, building society or credit union before the $250,000 FCS limit is then applied to the total for each individual. Case study

Alex and Peter have a number of accounts with their credit union:

$300,000 in a joint account
$50,000 in a separate account in only Alex’s name.

The FCS protects a total amount of deposits up to $250,000 for each account holder for each bank, building society and credit union. Therefore, Peter is covered under the FCS for $150,000 (half the joint account) while Alex is covered for $200,000 (being the sum of $150,000 from the joint account in addition to the $50,000 from his individual account).

That would mean that you need to split the funds across different financial institutions.

You have to be careful that the financial institutions aren't related:

The FCS protects deposits up to $250,000 per account holder at each separate bank, building society or credit union (also known as authorised deposit-taking institutions, or ADIs). So a person can have up to $250,000 in deposit accounts with a number of different ADIs, and they will all be protected by the FCS.

It is important to note that some ADIs market themselves under more than one brand or trading name. For example, BankWest is part of the Commonwealth Bank, while St George is part of Westpac. Some banks also offer accounts under the name of a different company, such as a subsidiary of the ADI: for example, deposit accounts offered by RAMS are actually Westpac accounts. Some accounts may also be branded or marketed under the name of a third party, such as Bank of Queensland offering accounts under “Virgin Money Australia” name or National Australia Bank offering accounts under the “Citi” name.

So, a depositor might think they have accounts with two different banks, when both accounts are actually with the same institution. More information is available on our Different banking businesses under one banking licence page.

united-states

Many brokerages allow you to sweep funds into FDIC backed account. That account though, is spread over multiple bank deposits.

Fidelity's FDIC-Insured Deposit Sweep Program details

In utilizing the Program, your uninvested cash balance is swept to a program bank where the deposit is eligible for FDIC insurance. If you have more than $245,000 in uninvested cash in your account, the Program will maximize your eligibility for FDIC insurance by allocating uninvested cash across multiple program banks. Assuming all the banks have available capacity, a customer could have up to $4 million of uninvested cash in their Fidelity Cash Management and IRA accounts covered by FDIC insurance.

united-kingdom

There is a very similar scheme in the UK. There is one big risk: If your bank cannot pay, then another bank or insurance can likely not pay either. You can open savings accounts in enough different banks to reduce the risk, you can't make it go away.