Office 365 Shared Calendar Without Mailbox. You could create a public calendar. Yes, they can either do a shared mailbox or a room mailbox normally used for things like conference rooms but great for shared calendars.
The first one is sharing a calendar in outlook on the web (owa), the second one is. To set up a common, shared calendar or contacts list that people in your organization can access and edit, you simply need to create a shared mailbox.