Yes sauerkraut can and does eventually go bad in the fridge.
When sauerkraut is kept in the fridge or refrigerator the sauerkraut will last for up to 6 months.
Sometimes the sauerkraut may last a few months longer than 6 months in the refrigerator as long as the sauerkraut container remains tightly sealed and the refrigerator temperature remains constant as well.
It's best to eat the sauerkraut within 6 months though and then have fresh sauerkraut for later.
The sauerkraut is not something you want to keep for too long.
But as long as you keep the sauerkraut sealed and refrigerated it should be good for at least 6 months.
If the sauerkraut has any mold in it or the sauerkraut smells bad then throw it out to avoid getting sick.