Medieval Ireland was not an AnCap society. It was hierarchically structured. Peasants paid obligatory tithes (taxes) but were protected by their lord or king, and other posts have already noted that slavery was present. Everybody in the society had a rank:
Early Irish society is hierarchical and inegalitarian. These characteristics are reflected clearly in the laws. So, an offence against a person of high rank entails a greater penalty than the same offence against a person of lower rank. Similarly the oath of a person of high rank automatically outweighs that of a person of lower rank. Native Irish law never subscribed to the Roman principle of all citizens being equal before the law.
Source:
The video contains some truths mixed with misleading simplifications. The English weren't the first to conquer, but they were the last. Previous conquistadors were subsumed into the existing society - the phrase "more Irish than the Irish themselves" refers to this phenomenon. This "AnCap" society was not able to defend itself.
As the video says, smaller petty kingdoms were slowly amalgamated into larger, each with an "over-king". However the entire island was ruled by a
High King (who also commanded taxes) - though much of the historical record is lost to legend by now. There was a highly complex legal system to which obedience was not optional.
For example, the video suggests people could easily change from one tuath to another. Not so; members of a tuath were unquestionably subjects of their lord or king.
The law texts refer to various types of outsider, and the distinctions between them are not always clear. There are many references to the ambue, the literal meaning of which seems to be 'non-person'. Hepdat 16 states that it is not a legal offence to avoid payment of a body-fine for an ambue. This would mean that an ambue can be killed or injured with impunity, so it is clear that this type of outsider has not come from a tuath with which there is a treaty. ... [The ambue] is thus excluded from normal legal agreements and remedies.
To summarize, there are some truths in what the video states, but the real truth is much more complicated - I'm certain it wasn't the AnCap paradise you're thinking of. Though, on reflection, my own interpretation of AnCap and, to a large extent, libertarianism, is that people with more money will be able to afford better "justice" than those without. So, actually, maybe you're right. It's not what you're thinking I'm sure, but I'll bet AnCap would end up fairly similar to medieval Ireland, only with guns and telecomms instead of swords and fast horses. The video admits that there were wars between petty kingdoms, calling them "minor things". I have a problem with this as an example of how an AnCap society can go wrong - when two "defence" contractors and their respective "mediators" cannot come to a satisfactory solution, the only valid response in war (given the absence of a society-wide independent judiciary - though see below). Notably ever since Ireland became a unified independent republic there have been no more wars within or across its borders, with the sole exception of terrorism.
One thing that interests me most is that the video states the king as also subjected to an independent judiciary - this is, to my knowledge, true (with the caveat that a king's word held more weight than a peasant's). I haven't finished the book I've quoted (it's a tough read), but I'd like to learn more about the judiciary's independence. One of the big failings of AnCap/libertarianism, in my opinion, is that it cannot guarantee an independent judiciary.