Most full e-mail clients keep a full locally cached copy of the server messages that can be backed up as needed.
Google will offer you many ways to backup mail on an IMAP based server:
https://www.google.com/#q=backup+imap+email
This article is specifically about doing backups of server based folders in Outlook:
https://support.office.com/en-za/art...c-aa1c33b18833
You can also backup IMAP mail by copying them to another server with another e-mail account in an email client. I find making a copy to Gmail very useful, as Gmail has good tools to use in searching and managing mail.
One note: I would not keep all messages in the Inbox forever. Moving messages to folders makes them easier to manage.