I feel terrible about my life at such a young age despite everything being in place for me.
Wow. Why do you feel so hopeless?
I went to school, finished my studies and I got a good part time job.  Yet everything is so unfulfilling and worthless to me.
Feeling terrible despite things being in place is something you have and need to deal with. I think a counselor will help you, go and make an appointment to see one.
I'm not ready to admit that something is wrong with me.
I think you already did admit it. You don't need to suffer in silence, it can be really hurtful. You should never feel ashamed for admitting you need help.