I specified in the the first sentence that you can't do it without involving another currency.
The government is the insurer of last resort when it comes to banks... for example the FDIC in the US.
I would never deposit money in a bank that guaranteed my funds on the condition I didn't mind being paid back in Rubles.
Since its impossible to have an insurer of last resort for bitcoin the bank would have to work in reverse.
Not true. You can have your USD denominated cashflow insured with USD by putting some of that USD into a pool. If your cashflow is interrupted, you take your share out of the pool that everyone has been putting into. That's how life insurance works.
With a Bitcoin bank, a bank can self-insure individual accounts by charging a small fee. If someone's account is lost, the bank can restore the money from the collected pool of fees. In the case of insuring an entire bank, there could be an insurance company that many banks pay a fee to (charged from individual account holders), and if the entire bank collapses, the insurance pool of money should have enough set aside to pay the people back. Best case scenario would be banks who insure a portion of their funds among many insurance companies, so if one insurance provider collapses, at least hopefully the remaining ones will restore some of the money.
IMO insurances are the biggest legal scams ever.
Essentially insurances insure you for something that is very unlikely to happen,
That, for starters, is quite illogical.
If something is very unlikely to happen, that is usually reflected in the price you pay. The chances of you getting killed or losing a limb in an accident are extremely tiny (compared to dying of health issues), and so AD&D (accidental dismemberment) insurance is only about $2 to $4 a month. The chances of you needing some sort of really expensive healthcare service when you are 80 is rather high, thus insurance is high.
For exemple, if an actuary assesses that your risk is extremely very low, then why the hell do you need an insurance in the first place?
You could cover that risk by yourself.
You could actually deposit a portion of your income in a jar (or invest it) and have yourself insured for when "shit happens" and there would be plenty of money available when it happens, since the probability of happening is quite low. If you collect your own money for each time that shit doesn't happen and use it when "shit happens", you should amass a considerable amount of money in your little jar.
You know how bitcoin hashing works. Accidents work the same way. Your chance of getting into a car accident that costs you $50,000 is extremely low, but it could happen today. Obviously the BEST method is to both pay for insurance and save up money. Specifically, when you buy insurance, yo pay for a certain amount of coverage, say $50,000. As you save up $10,000, lower your coverage to $40,000, and so on, until you are self insured.
In some cases, though, it is much better to buy insurance than to self-insure. For example, if I'm running a business that needs $100k in liability insurance, but likely will never use it, I would much rather pay a small monthly insurance payment and let the professionals work things out should anything happen, while using the $100k to actually do business and make money, than tying up the $100k in some untouchable bank account where it sits doing nothing, and then looking for lawyers and help on my own when something happens.
Because of this I still don't understand the compulsory nature of certain insurances.
If you have a freaking low risk, you should be exempted from having an insurance.
The compulsory part is because "freaking low risk" does not mean 0 risk. Low risk people still get into accidents, get infections, and even get cancer, yet because they thought they are low risk and didn't buy insurance, when they go to the hospital and get treated, everyone else who actually DID buy insurance ends up paying for them in higher insurance costs. Options are either force them to buy insurance too (and pay much lower premiums), or force hospitals not to treat them, and send them off to some island to die so the don't bother anyone here. The gov decided t go with option 1.
If this isn't enough: If you are insured and shit happens, be ready to be rejected because of a definition in cryptic minute clause in the corner of the backpage of the insurance policy that technically excludes you from certain conditions from collecting the insurance.
If that isn't conning I don't know what it is.
That's called "free market" and "read the damn contract"
Anyhoo, a bank that insures their money from being stolen in the very high risk environment like the online medium would be a freaking suicide, unless... you have placed extremely safe measures and extremely reliable redundancy to make attacks and data loss extremely unlikely, otherwise offering insurances would be totally senseless and stupid, business-wise.
Obviously I imagine that the fees or the premium for such warranty would be quite expensive and the investment for such infrastructure and personnel would be quite considerable.
That's true. So, that just means that the bank will have to weigh it's expense options between setting up extremely good security, or pay extremely high insurance premiums. Likely, it'll do what everyone does, and balance the costs somewhere in the middle (though since securing digital things is much easier than physical things like in a bank, I suspect they will exhaust every digital security option before going on to buy insurance.)