Yeah but they’ve still got your emails.
With your own Linux instance, you can host whatever you want, have full control, can host other services too like www, git, whatever, and have the assurance that you're not going to suddenly lose access because AI-BOT-204432 decided you violated some obscure terms of service. I've been doing this for close to a decade now (exim + dovecot for E-mail), and it works great. Back in the 90's, this used to be the default. How did we end up in this world where we so utterly rely on 3rd parties for such everyday critical Internet services?
The same way we ended up in a world where we so utterly rely on 3rd parties for such everyday critical services as growing our food and fixing our cars. There's too many things to do for everyone to do them all on their own.
For email? Because Google and Microsoft broke SMTP federation in the name of "anti-spam".
It's practically impossible to get Google, especially, to reliably deliver your email anymore if you aren't an actual email service provider.
I like GMail for their spam filtering power, and I honestly believe spam to be email's biggest weakness, and the reason people don't host their own. It certainly scares me, the thought of being flooded with thousands of spam emails daily, or the chance that my own emails would be falsely marked as spam since I'm not part of the major providers or because I did not configure it correctly. Don't know how this can be solved though, email itself is too permissive, and too "tweakable".
You can export all your email in a single .mbox file.
This might not work if your account is suspended, but if you set up email forwarding to an alternative address (e.g. to protonmail) that might still stay active so you can transition your addresses.
It also depends on the reason for the block - if for example they suspect you of having illegal content (child porn) in your Gmail, you aren't allowed to takeout it.
A balancing of rights exercise would need to be conducted by the controller to balance your right of access your personal data as against the identified risk to the third party that may be brought about by the disclosure of the information. The GDPR notes that these considerations should not result simply in a refusal to provide all relevant information, but the controller should endeavour to comply with the request insofar as possible whilst also ensuring adequate protection for the rights and freedoms of others.
I have very little hope indeed that they will let you do a takeout without finding a human to talk to when your account is locked.
For what it's worth, Facebook does let you do this. You login, get a message your account was banned for no apparent reason, and that you can download a copy of your data. Unfortunately it's broken (screenshot: https://dro.pm/a.png) but hey, there was an attempt.