It's not possible. Being a bank means negotiating individual contracts with a lot of other banks and the government central bank to let you transfer money. Banks aren't banks unless they have accounts at other banks.
Being a bank is a bit like being an Internet Service Provider: your job is to connect customers to a network, but there isn't a "magic internet box" somewhere which all the ISPs plug into. Instead, the ISPs plug into each other, haphazardly, and they are the internet. The only way to become part of the internet is to plug into someone else on the internet; ideally many people. It's the same way with banks.
When you send money from America to someone in, say, Australia, your bank can't just send your US$100 directly to the Australian bank - it has to figure out a chain of banks that have accounts at each other, and then phone them up and get them to transfer the money one step at a time. That's why it takes a week and costs $100.
In cryptocurrency, there is a "magic network box" that every "bank" plugs into. That "magic box" is called a blockchain. It's not really a "magic box", because inside the box is a network, that you have to plug into, but the process for running that network is fully automated and doesn't require you to sign any contracts or designate a compliance officer. You can run some pre-written software that connects you to the network, and then you can treat your copy of the software as if it was a "magic network box" and plug all your banking stuff into it.
But Bitcoin isn't very useful in the real world at the moment.
There's also the idea of a CBDC, or Central Bank Digital Currency, which would move one of the world major currencies onto a technology similar to Bitcoin. Unlike Bitcoin it would be a network with a central core in a highly guarded server room at the Federal Reserve. Banks would connect to the network core, using the Internet, and you would also have the option to connect to the network core, using the Internet, just like banks do. This technology is currently not deployed, and might never be.
With a CBDC system, you would send money to Australia by getting your Australian friend to make an account on the network, then sending $100 to his account, just like in Bitcoin. If he wants to convert it to Australian dollars, that's his problem, but he can just send it to some exchange company and get money at his other Australian CBDC account. Only one intermediate company is needed, and that's because of the currency conversion. No chains.
Or you can choose to only deal with the cash in your wallet.
