The obvious answer to your subject heading question is, you can't. Emancipation is not a right - it is a privilege granted to those teens who meet certain very rigid conditions and who have proven their ability to manage their own affairs. If you are not living on your own and managing your own affairs, you can't get emancipated. Simple as that. The law doesn't care how much you want it - if you don't meet the requirements, you don't get the privilege.
IF your parents allow you to move out AND IF an older sibling is willing to co-sign with you AND IF you can find a landlord willing to rent to you under those conditions, then you can move out. If your parents do not allow you to move out or if an older sibling is not willing to co-sign with you or if you can't find a landlord who will rent to you, then you don't move out. And if not moving out means you don't get emancipated, then you don't get emancipated.
Please do not post back with all the reasons why your home life is unbearable and you HAVE to move out because it will be so much better for you. There is no appeal. Either you qualify or you don't. Most teens don't.