1. In one sense, every Christian (in my opinion) must inevitably be at least a sympathizer with anarchism -- not in the sense of accepting its theory, but in the sense of considering every state in a relative light. No government does, or can, have the ultimate authority for a believer; in the last analysis, authority belongs to God, and government of whatever kind can go hang if it conflicts with our loyalty to Him. However patriotic a Christian may be, the state is not our purpose, and in fact every state will cease to exist, but no human being will: we are made in the image of God, but the state is not.
The refusal of the primitive Church to say that "Caesar is Lord" may be significant here. Given that Jesus is Lord, that He is our King, the idea of a competing king could never (for a believer) be more than a pragmatic concession in how we live, or so it seems to me. We believe in a monarchical cosmos, and the Throne of that cosmos is occupied. Hence, an earthly approximation of the celestial government will not necessarily be hierarchical the same way Heaven is hierarchical, precisely because Heaven is hierarchical. Humanity, having its rung on the ladder of being, may all be on basically the same level, from at least one perspective.
2. Related to this, I would point out that -- while, insofar as politics is based on a philosophy of man (particularly morals), and the Church does speak directly to that basis of politics, as she has a right and a duty to do -- the Church does not identify any political system as the correct one. She was born under an empire, continued through the collapse of the ancient world, helped build and survived the Medieval monarchies, and has weathered every modern form of governance from democracy to totalitarianism, without ever giving her unqualified approval to any specific setup of society.*
3. The most basic case for anarchy, in my opinion, was put by G. K. Chesterton (though he repudiated the term with contempt, and understandably, considering the character and arguments of much anarchism):
"This is the first principle of democracy: that the essential things in men are the things they hold in common, not the things they hold separately. And the second principle is merely this: that the political instinct or desire is one of these things which they hold in common. ... It is not something analogous to playing the church organ, painting on vellum, discovering the North Pole (that insidious habit), looping the loop, being Astronomer Royal, and so on. For these things we do not wish a man to do at all unless he does them well. It is, on the contrary, a thing analogous to writing one's own love-letters or blowing one's own nose. These things we want a man to do for himself, even if he does them badly. ... In short, the democratic faith is this: that the most terribly important things must be left to ordinary men themselves -- the mating of the sexes, the rearing of the young, the laws of the state."**
Taken with full rigor, this notion of self-governance is what anarchism proposes, at least in certain forms, and certainly in the form that I espouse. Philosophically, I ground this in the belief that all men, whatever their differences, are rational animals,*** and therefore have in principle something to contribute to the society of which they are a part, meaning that they must have a voice with which to make their contribution; this is strengthened by my Catholic conviction that all men are made in the image of God, and that it is this that is the final source of his political rights, rather than such rights springing from political expertise, or from some intrinsic superiority of one sort of person over another -- still less that they are conferred by the generosity of the state.
4. Jesus' own attitude toward the state in the Gospels seems to be one of marked indifference. Now, I will not make of His teaching an anarchist manifesto: He is my Lord, not my flag. In drawing out what I take to be anarchist implications in certain of His sayings, I am not making this a necessary part of my religion -- let alone refusing to be shown that I'm wrong.
Our Lord does not deny the existence of the state, or preach against it; yet He does seem to ignore it to the point of oblivion. In the narratives of the Passion, there is only one figure to whom Christ never utters a word: Herod. Pilate was the real political power, and He speaks to Pilate; the Sanhedrin were the motive force of the conviction, and He speaks to them. The people with power, the ones who can do something, He engages; for power, whether held by right or without it, is a reality of life. But the head of state, the theoretical authority, He ignores. The saying about rendering unto Caesar seems to sum the matter up neatly. Give Caesar what is his, and God what is His; but what isn't God's? Caesar wants money -- well, it has his image stamped on it, let him have it for all the good it'll do him. And human beings have the Divine image stamped on us: let Him have us, for all the good He would do to us. Christ seems, in every way, to regard the state and its operations as unworthy of the attention that the people were paying it; quite a disappointment to the hope of many Jews at that time for a successor to the Maccabees.
Pictured: The King of the universe, and also some dude wearing a crown.
(Christ Before Herod, Duccio di Buoninsegna, ca. 1310)
This attitude can be reconciled to many approaches to government, statist and non-statist; it is not a strong argument for anarchy. Yet -- Let this mind be in you, which was also in Christ Jesus ... Our vocation is, obviously, different from His; yet it is His outlook, His priorities, His values, which must shape ours. Whatever our calling and whatever our views, we must interpret them in the light of what He said was important, if we are to mean anything whatever by calling ourselves His disciples. And Jesus' whole attention lay upon the kingdom of God, not the kingdom of man. Even rebelling against the kingdom of man, even supposing it were a good end in itself, would have been a distraction from the task at hand. Whether anarchist politics are correct or not, the shift of focus that anarchism implies -- upon actual people, and not temporary abstractions like nations -- is right.
For heathen heart that puts her trust
In reeking tube and iron shard,
All valiant dust that builds on dust,
And guarding calls, not Thee to guard,
For frantic boast and foolish word --
Thy mercy on Thy People, Lord!
-- Rudyard Kipling, Recessional ll. 25-30
*The Catholic Church has straightforwardly rejected certain approaches to civil society, such as Communism -- partly on the basis of its outrages of religious liberty, and partly owing to its philosophically flawed theory of man. But a rejection of some specific idea as wrong does not in itself constitute patronage of any other idea.
**Orthodoxy, ch. IV: The Ethics of Elfland, pp. 43-44.
***Admittedly an optimistic theory. But men are at any rate rational or at worst irrational animals -- they are not beneath reason (in the Aristotelian/Thomist sense) as most animals are.