If you don't receive your salary paid in bitcoins and if you can't use your salary everywhere you will still need to use a bank.
Otherwise how do you plan to receive your monthly cheque or how do you plan to pay the recurring payments to pay the bills (water, electricity, ISP, phone, insurances, taxes, landlord...)? If you can not pay the groceries how will you pay the cashiers?
As a mater of fact this can be done with ease.
Some of my friends work in companies that pay them in cash. There's a dedicated office in the building where you go and get cash every month, pretty normal stuff.
Most contractors like gardeners, electricians, construction workers receive cash. Nobody expects them to cary around a card terminal and checkbooks aren't a thing in the EU.
Every bill can be paid in my local post office. You just bring them all your bills, they scan the barcodes and you're done.
It's possible to live without banks. Is it convenient? No, but that's a different question.