I am not an anarchist, so it is perhaps not relevant for me to discuss this: but I do think there are various forms of anarchism, which are at least on the table. In the past the term was usually associated with the left, and the idea was caricatured (notably by cartoons of men in black capes and big hats, carrying spheres marked "bomb", with a sparking fuse). More recently it has been associated with the right, and the "anarcho-capitalists".
From my point of view they have some things in common and some things which make them very different indeed.
Left wing anarchists seem to envisage a situation in which there is no centralised authority beyond that which is directly granted for a specific purpose: so those who say there is no order in such a system seem to me to misunderstand the proposition. The story of Cinncinatus, whether apocryphal or not, is a better illustration of what is implied here, perhaps.
As was mentioned, direct democracy is one way of achieving this. However this does not seem to me to be practical unless we return to city states (and even then existing cities are too big).
The devolution of power to the smallest possible unit has its attractions, and if such units cooperated to achieve those things best done by the efforts of a much bigger population then this might be a desirable route. This is maybe what was meant when an earlier poster talked about the global situation as an example: and it is not impossible to envisage this or to achieve it. It is certainly a thread in mainstream culture which often bewails the remoteness of government and the political class, so there is at least some political will in this direction.
As was mentioned in the "Overpopulation" thread there is no evidence that holding much in common leads to tragedy: but it does arguably lead to slower technological progress, for example. I am not convinced that is an unmitigated evil because it seems to me that rapidity leads to big mistakes sometimes - for example we got rid of trams in major cities in the uk and now we have to spend a lot of money to get them back: and the same goes for railways. More considered actions might have avoided some of those errors (or perhaps not: I am not committed either way though I think it is worth discussing.) At any rate the idea of certain things being part of the common rights of all within a community is not ridiculous, and it cannot be ruled out by appeals to human nature nor to common sense: those terms are not very helpful except insofar as they mark the philosophical stance of the speaker.
Right wing anarchists also advocate governmental authority limited to particular purposes (if they admit of the desirability of any at all). These are more akin to the characterisation of some of the posters here, because they can at least sometimes agree that it will result in adverse consequences for many people if there is no regulation. This is a fundamentally different conception of "human nature". The anarcho-capitalist is comfortable with the outcome, whatever it may be, because his guiding principle is freedom, no matter who that hurts. I am aware that there are those who expect the proposition to lead to a better life for all: the point I am making is that if it so happened it did not, they would still support it on principle: and the anarcho capitalist is happy with a very unequal society where the left wing anarchist is not.
For the anarcho-capitalist there is the problem that many of them do believe in property rights: there are those who would enforce those using private armies and police forces, and this is a way of attempting to resolve what seems to me to be an inherent contradiction in the position: but those are at least consistent. For the rest, there is often an acknowledgement that some residual power must be reserved for the state: and in that case there is nothing principled left in the position: they are really saying that "my freedom matters more than your starvation" and I cannot see that is much more than kleptocracy. Others will of course disagree.
The OP also seemed to be interested in other ways of organising society which we might be able to envisage. So I will put forward my most radical idea for the amusement of others. What I propose is that we stick with the units we have now or something close to them: and we continue to have representative democracy (because of size): but that the government is appointed on the jury system for a fixed term: and at the end of that term each member must return to the position he or she held before they served.
I do not see that politicians have special skills except those required to win an election: those would be redundant under this system.
The advantages I see are:
1. the governing body would be far more truly representative then it is at present
2. the members would be closer to the position of Rawls' "Veil of Ignorance", in that they would know they were returning to their previous position and would therefore have an interest in making sure that the situation in all levels of society was at least acceptable.
Far from anarchy, but at least another idea. I hope this is in line with the wishes of the OP