How can I stop OneDrive from blocking my emails due to storage issues?

You have control over what goes into OneDrive, so it's a tough choice. You'd either need to be super selective about what you back up or, as you mentioned, shell out for more space. If you didn’t choose OneDrive and it's just kind of there, that really sucks. At this point, looking for a different backup solution might also be worth considering!

It’s frustrating for sure, and I totally get where you're coming from. Sadly, Microsoft doesn’t separate email storage from OneDrive storage, which can really bite us in situations like yours. Your main options are either upgrading your subscription for extra space or deleting some stuff from OneDrive to make room. I know it's not the best solution, but it might help you regain access to your emails.