You will almost certainly find that there is an option provided by toucansurf.com to tell it to retain your emails on its server, as well as download them to Outlook. Right now, you have probably selected the option that says "once they're downloaded, delete them from the server". If you can do that, then you can log onto your email account from any PC. The other option is to sign up for something like Teamviewer or GoToMyPC which will enable you to view your desktop - and thus your Outlook files - from anywhere.
Surface Studio vs iMac 5K comparison review: Apple and Microsoft all-in-one PCs battle it out