If you are in an Airport, an internet Cafe or simply using another computer temporarily then you can still access your full Office 365 mailbox, contacts and calendar using just a standard internet browser such as Firefox, Google Chrome, Safari or Microsoft Edge.
To login to your Outlook Web Access service simply go to http://portal.office.com
Login with your email address and password
Click on the "Mail" tile
The Web Based Version of your Mailbox opens and can be used just like Desktop Outlook