I created an App with a few different pages. But when i navigate from one page to another - for example with a bottom Navigation bar - all the data the user can add or change on a page is deleted although I used Stateful Widgets. Do I need firebase to safe data or do Stateful Widgets safe data on my device?
In Flutter, no data is saved unless you use some kind of persistence mechanism - local or cloud storage/database.
A StatefulWidget does not persist your state even when navigating between pages, that's not the purpose of it - StatefulWidget simply provides methods to re-render the widget explicitly (basically that's the main difference between Stateless and Stateful widgets).
To answer your question, if you need to persist the data, you should use either local data storage solutions (shared_preferences, hive, sqflite etc.) or any back-end/cloud solution (Firestore, AWS Amplify, your custom back-end).