With the release of Office 2013 and it's emphasis on OneDrive storage, I've had several people ask about moving their pst file to OneDrive, DropBox and similar services.
The answer: It won't work in most services and is not recommended in any service. Outlook puts a lock on the pst file when the pst file is open. OneDrive (and other cloud solutions) continually syncs the local folder. It won't be able to sync the pst because Outlook has a lock on it and as a result, the pst file could become corrupted and data loss occur.
DropBox does incremental backups (reducing the need to upload a large pst in full each time), however, you need to make sure you close Outlook and wait for all updates to be uploaded before shutting down the computer, otherwise accessing the pst from another computer could result in lost data or a corrupt message store. Always keep a backup copy of the pst. This is an unsupported scenario and Microsoft will not provide support for it.
While you could use OneDrive or another cloud service 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 shut the computer down before its completely uploaded then try to open it from another computer, the pst may be corrupt.
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.
When you use a backup service I highly recommend verifying the backup several days after setting it up, so your sure it's getting the pst files and also every few months to insure it's still working as expected.