How does a 16-year-old get emancipated?

A minor seeking emancipation shall file a petition for emancipation in the juvenile court in the county where the minor resides. The petition shall be signed and verified by the minor, and shall include:

(1) The minor's full name and birth date, and the county and state where the minor was born; (2) A certified copy of the minor's birth certificate; (3) The name and last known address of the minor's parents or guardian, and if no parent or guardian can be found, the name and address of the minor's nearest living relative residing within this state; (4) The minor's present address and length of residency at that address; (5) A declaration by the minor indicating that he or she has demonstrated the ability to manage his or her financial affairs; the minor may include any information he or she considers necessary to support the declaration; (6) A declaration by the minor indicating that he or she has the ability to manage his or her personal and social affairs; the minor may include any information he or she considers necessary to support the declaration; and (7) The names of adults who have personal knowledge of the minor's circumstances and believe that under those circumstances emancipation is in the best interest of the minor.

This can be a long and tedious process and it is by no means automatic. You should consult with an attorney in your area about all of this. If money is an issue, see if you qualify for Legal Aid or see if they can recommend someone to help you. Also, check if there is a law school nearby to where you live; they typically run free/low cost clinics that handle these type cases. Finally, you can contact the local Bar Association in your county; they may have a list of attorneys who will take your case "pro bono" (for free) or at least for a reduced fee based on your income/circumstances. 

Note:  If there is abuse of some kind going on in your home go to your school counselor or contact your state's Department of Social Services; they can help.