I've had several people ask about moving their pst file to OneDrive and similar cloud-based services.
Sorry, no can do. It won't work and could result in a corrupt data file.
Outlook puts a lock on the pst file when the pst file is open. OneDrive continually syncs the local folder. It won't be able to sync correctly when Outlook is open because Outlook has a lock on the data file.
While you could use cloud storage to store backup copies of your data files, its not recommended due to the large size of many pst files: uploading a large data file can take "forever". If you want your Outlook data in the cloud, use Office 365 or other hosted Exchange service, Hotmail/Outlook.com or an IMAP account. Office 365/Exchange will store all Outlook data online, while Outlook.com/Hotmail is email, calendar, and contacts only.
Note that Hotmail Tasks will sync only with Outlook 2013 when you use EAS. The Outlook Connector does not sync tasks. IMAP is email-only.
There are online backup services that can backup pst files when Outlook is open and incrementally backup the pst, so only changes are saved. These are safe to use because they are designed to work with Outlook data files. However, unlike backing up a file to OneDrive or DropBox, you can't download the pst file to another computer or when needed. You need to restore it using the backup application.
Update 1/2012: I tested SugarSync to sync the folder where the pst were stored. It won't sync the pst files, but is syncing the temp files Outlook creates. When I received notification that I was almost out of space, I used the SugarSync File Manager to clean out files I didn't need and empty the Deleted Files cache. It turns out that I had several GB of deleted files going back months. I permanently deleted the pst.tmp files to reclaim 2 GB.
Published August 9, 2012. Last updated on September 15, 2014.