Now, I’ve googled the problem, and it seems that you could use the DontDestroyOnLoad() method to keep it running, however, the game Im making is a multiplayer game, and the class i want to keep running does not use MonoBehaviour, so my question is how do i keep my network-class running when the level loads (from the start screen)
The way unity handles object and scene only allows to use MonoBehaviour to use DontDestroyOnLoad. If you do not want your class to be MB then you need to store the info into the PlayerPrefs or any other media (database, text file, binary file, xml or else) and then recreate the object in the new scene using those values. It is just creating a clone of the previous one.
If you only have one object, you could also use static class but I would recommend the previous approach.
Create a saving system which in one method will save all needed info from your object.
Then in the Start/Awake of a manager class in the next scene, create a new object and use the value to create a clone.