This page of the guide to Syberia: The World Before explains if you can manually save the game.
Syberia: The World Before saves your progress automatically. Autosaves happen when you end a chapter and start a new one, when you complete main and secondary objectives, when you unlock new achievements, and sometimes even after picking up or examining an object.
Unfortunately, you can't make manual saves, so you must rely solely on the automatic ones. You also cannot go back in the storyline in any way. If you want to replay a chapter, you have to start the game from the beginning.