I replied to a similar thread from the lemmy.world selfhost community here: lemmy.world/post/24406909/14550114

My reply reviews what concerns you might want to consider before running your own email server.

To address your post specifically: yes. Kind of.

You can purchase a domain from a registrar. You don’t really own that though. You have to renew your registration periodically so it’s more like renting than owning for those at the “consumer” level. Still, if you register a domain then you can manage its Domain Name Service (DNS) records. Email routing is handled by DNS Mail Transfer (MX) records.

Once you’ve got a domain name you need to decide if you want to selfhost or purchase email services that let you bring your own domain. Both are doable but bringing your domain to a service provider is likely the better path for you.

Tuta mail and Proton mail are popular providers that let you bring your own domain.