Because both Exchange and Gmail share a common protocol, IMAP, you can use a program using IMAP to connect the two services.
Finding the correct tool for this may be problematic, however. Outlook is not an ideal solution, as it only wants to sync with a single server, be it Exchange or Gmail. There are quite a few scripted solutions, using perl or python , however they can be intimidating and may take some work to get them operational.
Assuming that you are only making a backup of your current emails for your own use, you could set up a Gmail account to retrieve emails from your Exchange mailbox using POP3, a simpler precursor to IMAP using directions found on Gmail's website (I need a reputation over 10 to post more than two hyperlinks per answer). Or, you can add rules in Outlook to forward mail to a Gmail account as it arrives in Exchange. However, neither method will provide a complete solution, as they will not back up your sent emails.