To start, yes this is possible, but you will need to use an App-Only token to access the Exchange APIs to get this data. It is important to remember that your access token for the API dictates what permissions you have to access that API. Assuming you are not an Admin, if you get a User token and try to query someone else's calander, you will get a "403 - Forbidden" since you only have the permissions to read your own calendar.
On the other hand, you can request for your application to have Application Permissions (versus Delegated Permssions) which will give you high level access to query the Exchange API without the context of a signed-in user. You can see documentation on these permissions here at Table 4.
Note that for any application that requests these level of permissions, it will require admin consent before the app will be able to get access. From the same page as above:
Note The tables describing service app permissions do not include a
Requires admin consent column, as permissions for service apps always
require admin consent. For more information about service apps, see
Building service and daemon apps in Office 365.
Please check out the link in that quote, as it may really help you get started with developing a service application.
I hope that this helps!