No, it is not possible without using nominees. Some jurisdictions (IIRC Sweden, NL, Belgium) have corporate entities whose shareholders are private or quasi-private, but this is changing with public UBO registers. It's difficult and time consuming to get documents out of some jurisdictions, such as Bulgaria and Gibraltar (which, for now, is still EU but be mindful of Brexit).
The closest you can get is probably Switzerland, where shareholders of companies companies formed as AG/SA are not publicly disclosed. This does not apply to GmbH/Sàrl companies.
Banking shouldn't be a problem if you work with good nominees, who have good relationships with banks. Take Cyprus as an example. Your chances of opening a bank account in might even increase if you use professional directors and nominee shareholders. The drawback is you're limited to the banks your nominees work with.