Ask HN: Do you backup your Emails?

I wonder if its just me or other people feel the urge as well to backup their emails. I feel oddly attached to my older emails and have the fear of loosing something important which might hide in my mail archives.

Also I don't necessarily trust some of my email providers (nor my self hosted one) and soonish have to clear one out because I run out of storage.

So what tools are you using to backup your emails? Is there a service which takes care of this automatically?