How did that work in Eden? How did that work in Israel? It will one day work after Jesus returns, but the unfortunate reality of man's present fallen state is that our natures work against that.
...and as far as being free of all earthly authority, Jesus said that the gates of hell would not prevail agains His church in the Bible. Many misinterpret this verse to apologize for the Church of Rome, but in fact refers to the catholic (universal) church around the world.
Actually, Eden is the perfect example of Christian Anarchism. In Eden, and even immediately afterwards, Adam and his wife, Eve, were beholden strictly to God Himself, no other earthly authority of man.
In ancient Israel, originally the nation was communal, and agrarian. When the nation displeased God, God would send forth a Judge to set the nation straight. They didn't have an established monarchy or "government" as we know it. It wasn't until the time of Samuel that Israel began to demand a king like other nations around them. God warned them through the prophet saying:
1 Samuel 8:10-22 English Standard Version (ESV)
10 So Samuel told all the words of the Lord to the people who were asking for a king from him. 11 He said, "These will be the ways of the king who will reign over you: he will take your sons and appoint them to his chariots and to be his horsemen and to run before his chariots. 12 And he will appoint for himself commanders of thousands and commanders of fifties, and some to plow his ground and to reap his harvest, and to make his implements of war and the equipment of his chariots. 13 He will take your daughters to be perfumers and cooks and bakers. 14 He will take the best of your fields and vineyards and olive orchards and give them to his servants. 15 He will take the tenth of your grain and of your vineyards and give it to his officers and to his servants. 16 He will take your male servants and female servants and the best of your young men and your donkeys, and put them to his work. 17 He will take the tenth of your flocks, and you shall be his slaves. 18 And in that day you will cry out because of your king, whom you have chosen for yourselves, but the Lord will not answer you in that day.”
This clearly demonstrates that God was not "for" the nation of Israel having a king. In fact, God warned them against having a king. However we read...
1 Samuel 8:19-22 English Standard Version (ESV)
19 But the people refused to obey the voice of Samuel. And they said, “No! But there shall be a king over us, 20 that we also may be like all the nations, and that our king may judge us and go out before us and fight our battles.” 21 And when Samuel had heard all the words of the people, he repeated them in the ears of the Lord. 22 And the Lord said to Samuel, “Obey their voice and make them a king.” Samuel then said to the men of Israel, “Go every man to his city.”
We get God's general feelings on the matter when God states:
"Obey the voice of the people in all that they say to you, for they have not rejected you, but they have rejected me from being king over them."
God never intended Israel to be governed by a king other than Himself. The Christian Anarchist seeks to experience and realize this level of God led rulership, rejecting all earthly civil governments as being an ultimate authority over the Christian.