No, it’s rather unlikely. Especially if you use some DNS service that blocks some ICANN WHOIS lookups (usually gives out information that you enter when registering the domain name), you’ll be fine.
Domains can be cheap, but will need to be renewed. A VPS for hosting can be cheap or expensive, but running it on an older computer would be a good bet.
The benefit of a VPS is that you don’t need to worry about using your own IP address.
wouldn’t that allow me to get doxxed if I self host?
No, it’s rather unlikely. Especially if you use some DNS service that blocks some ICANN WHOIS lookups (usually gives out information that you enter when registering the domain name), you’ll be fine.
Domains can be cheap, but will need to be renewed. A VPS for hosting can be cheap or expensive, but running it on an older computer would be a good bet.
The benefit of a VPS is that you don’t need to worry about using your own IP address.