HUGE Outlook Database. HELP! – Using Exchange. Can I move folders to cloud service? Suggestions?

Hi - 

My wife is in kind of a pickle with her 12 years of Outlook folders, emails and files synced with Exchange.  The database file for her Outlook identity is approaching nearly 40GB of space and really causing a ton of issues.  Have had to rebuild, create new identity and loads Outlook incredibly slow. 

What I'm looking for is a suggestion of how to move some of these files to the cloud (storage - still searchable, accessible and usable) and out of her locally based identity folder.  Any thoughts?

What we have done:

• Locally based files.  This doesn't take any stress off the computer.
• Drag and Drop to Gmail.  We're talking about tons of folders here and it times out if the folder is too large.
• Leave them as they are.  Yup - this is how we are now.

Most of these folders and files are clients from 4+ years ago that she legally needs to keep for reference and if they want to buy or sell again (she's in real estate) she wants to revisit the emails.  

I keep coming back to the way that my iPad/iPhone work.  You can search particular folders and files but they are NOT stored on your phone/tablet unless you download them.  Is there a way to keep a "shadow folder" in Outlook where the folder content isn't locally stored but still searchable/accessible?

Thanks for your help!