Hey. I can completely relate to feeling so worthless. But I have to point out that while you are a part of your family, you are not directly responsible for your parents successes or failures. Seems like they are adults, just like you, so they have to be responsible for themselves. Yes, you all live together, so there is some shared familial responsibility there, but you cannot take the burden of your mom having to stay at a job she doesn't like.
Are you contributing to the bills?
Working is hard sometimes, most especially during depression and regardless of good intentions, sometimes its just impossible to find the motivation to do something, anything. Its part of the definition for depression. And fears are also hard to get past. Just saying that they have no purpose doesn't really make them go away. Most fears are irrational. Shi# most of the things I have problems with are completely illogical and I know that, but I am still struggling to make the tiniest increment of change. Sometimes just saying everything that's stressing you and bringing you down is helpful. Maybe getting it all out with help and you will be able to move forward right away. Maybe not. Either way, you are not worthless. Things will change. Maybe even for the better.