You can, sort of. It depends on how technically you define the term "bank" and how far you're prepared to stretch the term "self-hosting". In principle, you could start your own bank, but I understand that's not the intention.
Technically, building societies are not banks:
So, what exactly is the difference? Key to understanding the distinction between banks and building societies is to be clear on exactly how, and for who, they operate. Because banks are listed on the stock market, they are businesses and therefore work in the favour of those who invest in them, specifically their shareholders. Building societies, however, are not commercial businesses, they are ‘mutual institutions’ – owned by, and working for, their customers.
- What Is The Difference Between A Building Society And A Bank?
Since building societies are "owned by, and [work] for, their customers", you'd kinda-sorta be "self-hosting".
If instead you want to define "bank" more loosely and "self-hosting" more carefully, then what you want is not possible.
Hypothetically, suppose there was a service that you could subscribe to that issues a card to you. You would be able to deposit money into the platform, and the platform would have the appropriate business relationships to allow you to use your card with third-party merchants.
That hypothetical service would then be a bank of some sort, even if it was called a building society.