If you are receiving errors about being close to your quota for your Exchange e-mail account, you should consider archiving your older messages. Archiving copies old e-mail messages and attachments to an Archive.pst file (you can change the name during with the manual export process) and then removes the data from the Exchange mailbox decreasing its overall size. Items moved to an Archive.pst file are still accessible by opening the file through Outlook, and should be included in regular backups to prevent lost data. If you need more assistance in managing your mailbox size, see the Quota Management page for Outlook 2013.
The archived folder is now available from the Mail window, in the All Mail Folders section. (Note: the actual file is saved on your hard drive, not the exchange server! If you need to remove your account profile, you will need to import the file back into Outlook to gain access to the files again. Your .PST files should also be included in any regular backups you make.)