Yes, data and mapping can be done completely without internet and mobile network service.
Read more about Setting Up a Complete Project, including taking surveys offline, here. Scrolling down to the Sharing a Web Map for Offline Use section will specifically tell you how to set-up an offline survey’s map for offline use.
The videos on the Data Collection Training Videos and External Resources page will further guide you on how to work offline.
If you want to know more about what to do after you have collected data and mapped an area offline, take a look at Sending Surveys Collected Offline.