The Shire was not an anarchy nor did it have a government; but had a clan- (family-) based system of authority

Some people have badly misunderstood The Shire as some kind of an 'anarchy'; with the implication that a lack of much in the way of formal government mistakenly-equated with a condition of unconstrained individual 'freedom'. 

But this idea of the unconstrained individual, free to do whatever he wanted, is very modern; and anarchy did not exist in traditional societies - although the delusion that it did, and that ancient tribal hunter gatherer societies were some kind of anarchy dates back at least to Engels, and Rousseau before him. 

On the contrary; it seems clearly implied that The Shire was - like many ancient and small scale societies - based upon extended families - that it was a kind of clan-based society.

In clan societies, individuals 'identity' is as part of their family; and it is families who regulate individual behaviour - using the relatively fluid and informal, but nonetheless authoritative, means by which families spontaneously operate. 

Thus we are told that hobbits are deeply interested by genealogy - that is family ancestry; and individual hobbits are nearly always discussed in terms of their familial descent and membership. 

Furthermore; there is also an implicit Shire class system; with some hobbit families identified as 'gentry' (such as Baggins, Bolger, Boffin) - and a few of these (such as the Tooks primarily, and Brandybucks secondarily) as something-like an aristocracy in that they have an hereditary and patriarchal (primogeniture) authority over others. 

This dominance by certain clans appears to be rooted in the hereditary competence for rulership of Fallohide hobbits; who seem to be both bolder and more intelligent than the other Stoor and Harfoot types.  

(Although formally patriarchal; it is also clear that  - just as in real life Men's families - exceptional female-hobbits may establish substantial - but non-hereditary - authority, or even dominance: examples are, buy implication, Bilbo's 'remarkable' mother Belladonna Took, and Lobelia Sackville Baggins, who we hear described apparently dominating first her wealthy husband, and later her son.)     

In such a clan society; there is almost nothing corresponding to the individuality of anarchism; since personal identity is - spontaneously - immersed in the family; which is indeed regarded by other hobbits as definitive. 

Bilbo and Frodo, as wealthy (hence independent) bachelors, who care little for the opinion of other hobbits, are, of course, notable exceptions, since they - apparently - do whatever they like! 

But there will have been extremely few Shire hobbits who could afford to lose their 'respectability' (i.e. their guaranteed place in Shore society) by steeping outside of family expectations.  

An important factor is that The Shire does not much resemble any actual historical society of Men; and this is mainly because hobbit-nature is different from Man-nature - especially in that hobbits have very little desire to dominate and control others; so that Big Man societies or monarchies do not develop. 

Again, there is an exception with Lotho Sackville-Baggins, who is power-driven and briefly becomes a Big Man (the "Chief"). Lotho is widely disliked, yet meets with very little significant resistance to his monopolizing of power from other hobbits, confirming their innate lack of dominance instinct. 

The exception is the Thain - who is head of the Tooks, the most powerful clan; and with the strongest Fallohide inheritance which fits them for ruling both intellectually and temperamentally. Thus, the Tooks refuse to acknowledge Lotho's authority, and indeed defend their territory with lethal force when Lotho send 'ruffians' to intimidate them.

In conclusion, The Shire is very far from an anarchy; and instead seems to be a society with a clan-family structure that is similar to those of many historical Men, plus a small degree of national government (e.g. the mayor, shirrifs and postal service). But the end result is significantly modified compared with Men to become something unique; by the different and less dominating and power-driven nature of hobbits*. 

 

*Note: This resistance to the temptations of power is exactly what makes hobbits the best of all races to bear The One Ring; and indeed suggests that this is precisely the reason that Illuvitar enabled hobbits to 'evolve' from Men, probably during the Third Age.