An interesting question! It's true that no machinery of state coincided with the existence of a complicated legal system. In early medieval Ireland, kings had no recourse to police services, standing armies, established bureaucracies or forms of institutional taxation that we might associate with state societies. In contrast to a contemporary statesman who is legitimized and shored up by the innumerable apparatuses of the modern state, the Irish king drew his power and legitimacy from two sources: his ritualistic role as a sort of "priest-king" whose primary duty was to uphold the metaphysical well being of his kingdom and his personal power and influence, which was drawn from the same system of clientship and patronage I've described in your previous post.

Early Irish kingship was a pretty spiritual and cultish position that seems derived from pre-historic and obviously pre-Christian religious and political practices. Legal sources such as the Lebor na Cert and literary texts like the Destruction of Da Derga's Hostel depict a crucial element of kingship which betrays the office's religious origins; Irish kings were expected to abide by buada and geassa - taboos and prerogatives that kings were expected to uphold lest they break them and bring destruction and death upon themselves and their territories. These taboos depended on the title but could prevent them from allowing strife in their household, travelling clockwise around their fortress 3 times, maintaining fleets on a specific lake and so on. This is compounded by the legal expectation for a king to maintain a physically unblemished body (though there are no historical examples of kings being stripped of their office due to injury) and a cultural conception that the rule of a good king will usher in prosperity in crops, wild game, cattle and good weather, while the reign of a bad or wicked king will bring about illness, plague, unsuitable weather, poor harvests and death.

Related to this concept of a king as an embodiment of everything that's good with with kingdom is his legally ascribed role as judgement giver. Irish court procedure is basically unknown but we do know that kings were supposed to give righteous judgements in certain court cases, which actually one of the only "state-like" things that they were allowed to do. Kings who made bad judgements could bring about the same sorts of supernatural retribution. It's quite possible that in pre-historic times Irish kings were simply a symbolic and purely priestly figures whose sole duty was to uphold the metaphysical wellbeing of their territory. And I do mean THEIR territory literally - Irish kings are recorded to have symbolically married a female personification of their territory upon coronation at an obviously pre-Christian ritual that apparently featured animal sacrifice.

So this is where the second part of the equation comes into the mix: why exactly did kings stop becoming a purely ritualistic and ceremonial figure with no executive power (which is attested in legal texts but does not exactly correspond with the actions of kings in historical documents) to the warring regional warlords that we all know and love? It seems that this process began sometime around the fall of the Roman Empire as Irish lords enriched themselves by raiding the former province of Britain, and freed themselves from manual labour with the help of slave labour captured there. Then they were free to establish themselves and their households as mobile warrior-aristocrats who then hemmed in the lower classes into subservient client roles (which I've described in the other post) meant to sustain the aristocracy with food and labour.

This new aristocratic development of social relations became extended into the political sphere; kings were not merely just kings of their individual territory or tuath (sometimes poorly translated as "tribe"), but developed complicated patterns of overlordship between these individual territories. One king became lord and patron of a few of his neighbouring kings, but might then have been subdued by another more powerful king who we might identify as a provincial king (in reference to Ireland's 5 historic provinces). Such a king might have dozens of kings subservient to him as clients in a complicated web of personal relations. These relations were personal rather than institutional because a provincial king earned the subservience of less powerful regional kings through actual personal contracts, not because the king of so and so was legally subordinate to the provincial king of such and such (like how the Duke of Normandy was de jure subordinate to the King of France, I suppose).

So how did Irish kings rule despite existing in a more or less stateless society? Because they served a ritualistic and priestly role that had over time been overridden by a new kind of social, economic and political system: that of aristocratic clientship and patronage. Irish kings "ruled" despite not having a state because they drew upon their own personal relationships, which were really just and extension of the same kinds of relationships of power that existed on every level of society, and because the historical origin of their office enabled them to do so.