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.
They do now (although you only have 7 days to do so I think).
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.
The GDPR (in Article 15(4)) states that the right to obtain a copy of your personal data should not ‘adversely affect the rights or freedoms of others’. This means that when responding to an access request, the controller should consider the rights of third parties, such as their data protection rights, trade secrets, or intellectual property rights such as copyright. This could arise, for example, where your access request relates to a record containing both your personal data but also the personal data, trade secrets, or intellectual property of others.
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.
Google doesn't let you Takeout when you don't allow Google to track your browser. It will give an error to "please use a device you regularly use" even if you can see in your account that there are no other active sessions that you could possibly make use of. I tried a few hours later with the session still open but no dice. Guess I'll have to find a human to talk to at Google in order to get my data pursuant article 15, GDPR.
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.
https://takeout.google.com/
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.