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.
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.