You can have a hosted e-mail system with your own domain, many service providers will do this. They would have the storage, archive, etc. on their servers. Access would be a web client, e-mail client of your choice, or a bridge demon. Option 2 is a hosted vertual server, so your self hosting but renting a VM server on a providers data center, they take care of the hardware, power, backup (sometimes depending on service), etc. You would set up your OS and e-mail services.

I use a combo of this, I have a service with my own domain and point my MX records there, I use a bridge and a sctipt to pull backups and store them locally but use their email clients for mobile and Thunderbird on PCs.

Edit: FWIW I’ve been using Proton (paid version) for this but will be moving to hosted virtual server at some point, when I get a minute to migrate. I expect the cost to be about the same

I can’t see the letter anymore (the site seems to be down) but unless this is a published public API or they have a public “bug bounty” program and the API was following the rules of engagement, and it did not sound like either was true, then this is hacking and illegal, at least in the jusidiction I’m in.

The next step should be to press charges on openAI, unauthorized hacking into computer systems is illegal.