Calendar.app stuck on iCloud sync
Uncheck the calendar in system preferences -> Accounts (or delete the whole account, I don't think it makes a difference).

Then delete files associated with the Calendar
~/Library/Calendars
~/Library/Caches
~/Library/SavedApplicationState/com.apple.iCal.savedState

Reboot
Open calendar, check if it opens normally.
If it does, quit calendar, add the account (or tick for calendar syncing) again, and reopen calendar.

Perhaps use something like Remote Desktop, VNC or TeamViewer if you don't have physical access to the computer.