The answer depends upon what risk you wish to protect yourself from.
If that risk is: my 'free email' provider decides to shutdown my email with no notice, then the risk mitigation is: "buy a domain name".
Do note that "buy a domain name" and "host your own email server" are orthogonal. You can "buy a domain name" and pay someone somewhere to "host an email server".
If your risk is: my domain name provider will terminate my domain name arbitrarily, well then, you are getting into very expensive territory where you'd have to become your own registrar, buy a TLD, and so forth. But then you just moved the risk up to "my TLD provider might terminate me".
Eventually, you realize that there is no way you can mitigate all the possible risks (not without personally owning all the world wide internet infrastructure), so you stop worrying about the "what ifs" that are so remote that they will never happen.
The simplest, lowest cost, and most risk removal is simply: "Buy a domain name". You eliminate so much of the risk there that the remainder becomes "very unlikely to happen territory".
If you own the domain name, then you get to move it around to different servers (and, as email natively supports 'backup servers' you even get to have multiple servers hosting it, provided you want to pay for the expense of doing so). This mitigates all the risk of any given "server provider" dropping you. You just move the domain to another and you are back in business as if nothing happened.
Your second step to mitigate risk is to never, ever, leave the only copy of your email archives on any of the servers you pick to host the email. For all of them, as soon as they receive the email (or as soon as you poll for an email) you download it to your local machine that you control (and backup). That way, any given server closing shop tomorrow, or canceling you tomorrow, has zero impact on your ability to access your archives of past emails.
Once you take those two steps, the remaining risk possibilities become remote enough that you really need not worry about them.