Check all "Appointment" Type Project Activities to ensure that BOTH a Start date, and an End date have been entered on all Appointment Activities.
If the Project Activity Type was at some point changed from an 'Appointment' type, to a 'Task' type, this can cause an Error when Saving. This Error can only be resolved by deleting and recreating that specific Project Activity. If you know which Activity was changed, simply delete it and recreate. Otherwise, you'll either want to delete all the project activities and reenter them, or you can try replacing them one at a time and try to save each time until you delete the correct one.
If you have a large number of project activities, you can narrow down the records which are missing either the Start date or End date by creating a List View from the Project Activities Tab. Be sure to include the Project Activity Type, Start and End dates in the List View.