I'm creating a Sharepoint intranet solution in my site and I want to be able to give each user a personal library/list of they're files that only they can view/edit. How is this possible in Office 365 - SharePoint Online?
Office 365 is not different from On-Premise SharePoint Sandbox Environment from a Development point of view.
So while building the solution if you build with keeping the Sandbox restrictions in mind, you will be good to go.
For your particular problem, I would look into User Permissions. Allowing only the authorized users to view/edit the list.
Here is a link to get you started with the SharePoint Client Object Model and User Permissions: http://blogs.technet.com/b/speschka/archive/2009/11/01/using-the-sharepoint-2010-client-object-model-part-5.aspx