oh lol... I'm from Korea too and the same thing happened to our family as well. You are eligible. You can apply for deferred action after 60days, and this will allow us to apply for work permit (EAD card), and through the EAD card, you can apply for social security number, so using that number you can work anywhere you want. However, for now, try to wait for 60 days till the whole process is setting up and meanwhile, you can gather any kind of legal documentation (whether expired or not) that will prove your enterance date to USA, and document that proves you been here for five years (school transcripts, bank accounts, utility bill, etc...)