I second Howardb's suggestion. I have about 10 PCs on a home network, and any user can log onto any PC and get their email and docs.
For the last decade and a half I've inter alia done this - simplifed version to accomplish your goal):
1. In the user's My Documents folder, create a folder called Mail.
2. Move the PST file from the user's default Local Settings/AppData location to the Mail folder under My Documents. I rename the PST file to something more readily identifiable, like AlexMail.pst. (If you Copy, then rename the original PST file to something else, otherwise your Outlook will default to the original location and file).
3. Place the user's My Documents folder (together with all their docs and folders) on a network folder. Share that folder on the network - you can use Permissions and Security to restrict access/sharing/privileges to one or several users, each exactly tailored as you wish.
4. Point the local profile's My Documents to the network location, so that when the user clicks My Documents it opens the folder on the file server (which can be a peer on the small LAN). Do this by right-clicking My Documents folder, then Properties > Location.
5. Start Outlook. It will fail to find the default/original PST file (moved/renamed in 2 above); when prompted, repoint it to the PST file in new Mail folder created in 2 and 3 above.
6. Create identical user profiles on each PC, and make sure each user's My Documents and Outlook point to the same location/file. Configure each user's Outlook identically on each PC.
7. Voila! Any user can log on anywhere and get their docs and mail. You can't concurrently open the same PST file in two instances/PCs (it's locked, correctly), but at least the identical PST file is available to the both systems, and you can be sure it has all the emails, deletions, address book/contacts, calendar, etc, etc. The only things that don't come across with the PST file are junk mail settings, customised templates (but simply copy NormalEmail.dotm in O2007), and Rules.
This method has the added advantage that the PST file is backed up when My Documents is backed up (which I do daily for all users).
As you learn more about Permissions and Security you can add greater granularity in managing each user's access to files and folders.
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PS. Storing personal email in a cloud owned by the world's most voracious information-hoarder gives me the heebie-jeebies. It'll be there forever, and can/will one day be used in evidence against you.