You can renounce your citizenship, and leave the country.
The details can be found on the State Department's website and the law that governs it, (Section 349(a) of the Immigration and Nationality Act (8 U.S.C. 1481) but in brief:
(5) making a formal renunciation of nationality before a diplomatic or consular officer of the United States in a foreign state, in such form as may be prescribed by the Secretary of State; or
(6) making in the United States a formal written renunciation of nationality in such form as may be prescribed by, and before such officer as may be designated by, the Attorney General, whenever the United States shall be in a state of war and the Attorney General shall approve such renunciation as not contrary to the interests of national defense.
They also note that renunciation of citizenship won't in itself shield you from taxes, military service, or other financial obligations:
The U.S. Department of State and the U.S. Supreme Court have concluded that the intention to relinquish U.S. nationality required for purposes of finding loss of nationality under Section 349(a) of the INA does not exist where a renunciant claims a right to continue to reside in the United States, unless the renunciant demonstrates that residence will be as an alien documented properly under U.S. law. [...]
Persons considering renunciation should also be aware that the fact that they have renounced U.S. nationality may have no effect whatsoever on their U.S. tax or military service obligations. Nor will it allow them to escape possible prosecution for crimes which they may have committed in the United States, or repayment of financial obligations, such as child support payments, previously incurred in the United States or incurred as a United States national abroad. Questions about these matters should be directed to the government agency concerned.
This hypothetical person wants neither the protection of the state nor the services provided by it. [...] this person would wish not to do: pay taxes, invest in social security, purchase healthcare, obey the police.
Not obeying the police is pretty easy if you aren't in the US. They likely do not have jurisdiction outside the US. Failure to follow a lawful order is usually a criminal offense in most US states (I just used the first google link)
§ 20-114.1. Willful failure to obey law-enforcement or traffic-control officer; firemen as traffic-control officers; appointment, etc., of traffic-control officers.
(a) No person shall willfully fail or refuse to comply with any lawful order or direction of any law-enforcement officer or traffic-control officer invested by law with authority to direct, control or regulate traffic, which order or direction related to the control of traffic.
Purchasing healthcare isn't a requirement for non-citizens.
Undocumented immigrants aren’t subject to the individual shared responsibility requirement.
Non-citizens who do not work in/for the US do not pay Social Security or Income Tax. You could chose one of the professions that is exempt from these taxes and reside in the US, but that would require you obey police officers.
Nonresident aliens, in general, are also liable for Social Security/Medicare Taxes on wages paid to them for services performed by them in the United States, with certain exceptions based on their nonimmigrant status. The following classes of nonimmigrants and nonresident aliens are exempt from U.S. Social Security and Medicare taxes:
The Internal Revenue Code imposes the liability for Social Security and Medicare taxes on both the employer and the employee who earns income from wages in the United States. The Code grants an exemption from Social Security and Medicare taxes to nonimmigrant scholars, teachers, researchers, and trainees (including medical interns), physicians, au pairs, summer camp workers, and other non-students temporarily present in the United States in J-1, Q-1 or Q-2 status.
