There is︂ nothing illegal about having an offshore bank account. Even if you're from US it's perfectly︃ legal. What is illegal is hiding assets and tax evasion. The US requires a declaration︄ of your offshore assets. Australia and pretty much all countries except US actually, require a︅ declaration of taxable income. For countries that tax residents on worldwide income, like Oz, you︆ declare the income you earn offshore and pay tax on it. If you have a︇ bank account offshore that earns no interest and receives no income, you never need to︈ tell the ATO about it.
As for bank requirements, that's up to each individual bank.︉ Some won't accept non-residents. Some will.