If you want to maintain access to your e-mail for the long term, take some time to think through what you really want to keep and let go of the rest.
Then make a plan for keeping those kinds of e-mails you really need.
These links can help you make a plan!
Keeping Personal Electronic Mail (excerpted from the Library of Congress, 1.8.2017)
Identify all your e-mail sources
Decide which messages have long-term value
Export the selected messages
Organize the saved messages
Make copies and manage them in different places