You don't need to host the email servers yourself. Many email hosting services will let you use our own domain with them.
If you want to use an email hosting service that does not directly support using your own domain, many domain registrars include free email forwarding so you can forward mail sent to your domain to your address at your email host, although there might be problems with sending from your domain if you use the forwarding approach [note 1].
It might at first seem that this is just pushing the problem back a little. Instead of the problem being losing your account at a mail hosting service like Gmail, you now have to worry about losing your domain.
The big difference is that a domain registration is a lot more passive. With a Gmail or other mail hosting account it is something you are actively using. Content you generate goes through it. Content other people generate goes through it to you. That gives all kinds of opportunities to trigger false positives on their automated anti-abuse systems.
With a domain you register it and designate name servers and periodically pay to keep it from expiring. Most registrars include basic name service so you don't have to deal with finding a name service provider. Once you've set up name service to designate your email host as handling your domain, or set up forwarding if that's what you are doing, you pretty much don't have to touch anything there and content to/from you doesn't go through those systems so there is simply much less opportunity for something to trigger some sort of automated anti-abuse systems.
Pick you domain and registrar carefully. Don't pick a domain name that is close to some trademark. Pick a top level of .com or .net or maybe your country's top level if you are going to want to send email from that domain [note 2]. Pick a registrar that is not in some country likely to do things that get your country to put sanctions on it.
[note 1] You might not have enough control over the headers on outgoing mail to be able to send a mail that doesn't look like a forgery attempt. For email addresses that you will just need for receiving things the forward approach should be fine, which will cover email needed for account recovery in most cases.
[note 2] The newer top level domains that are available for general use have been pounced upon by large numbers of spammers, to the point that having an email address in them can make it very difficult to get through spam filters. Spammers are all over .com and .net too of course, but that's also where most of the non-spammers are too. With the newer top levels the spammers jump on in large numbers from the start and so from the point of view of a random email receiver those domains are mostly spam.
