How should I go about getting a place of my own as an emancipated minor?

If you have been legally emancipated, you should take the order with you when you apply for an apartment, housing, etc.. One of the difficulties you have in your situation is that folks may not want to rent to you if you are under 18, in spite of that order. There is nothing illegal about someone refusing to rent to you because of that. Hope fully you have persons in your life who can help you with this question, as it is mostly a practical life question as opposed to a legal one.