I managed to get through the first days at relative ease. After several attempts, I can't earn enough money on the day I'm at to feed and medicate my family. Is a stalemate situation possible?
First of all, it is entirely possible to play the game without making any mistakes. See this video (I didn't watch it, but it claims to be a perfect playthrough).
As Matthias noted in a comment, you are not expected to be able to always play from the latest day and get further. There are some situations where because of a past action, you will be arrested, or your family will die, or some other ending. You are encouraged to continue from a previous day or begin a new game and attempt to reach a different ending.