Feudal, but women have a great deal more power, and freedom. Also, not entirely middle aged, as they have computers, television, and cinema. So we can expect far more in the way of womens liberation, which means less arranged marriage.
The more she learned about the Elemental Nations the more convinced she was that her path of escaping the system was the only one that she would accept. The Hidden Villages lived in a mutually dependent relationship with the nobility in their home countries, with the Daimyo gifting a large percentage of their economy directly to the Hidden Villages in order to keep them loyal and to help maintain standing armies. The nobility themselves were a corrupt and cruel bunch who in many ways behaved almost like caricatures of an oppressive ruling class towards their subjects. And why wouldn't they? Any peasant uprising would be quickly quelled by the ninja and the nobility's own private guards would keep them safe from any retaliation.
This looks like its implying a feudal system, though in practice thanks to the power of the Ninja, I wouldnt be surprised if the Elemental Nations were essentially a slave society.
Its not an unsurprising thing outcome based off of chakra. Ninjas have to devote large portions of their time to training, and ninja stuff, rather than learning how to govern, get crops, manage stuff. So they get the nobility to do manage it for them, and share the rents they accumulate with them. The Nobility have no incentive to treat the serfs who do the work well because they have no fear of peasant rebellions thanks to ninjas. Thus they can brutalize the peasantry as much and as efficiently as possible and essentially obtain maximum effort and productivity from every serf.
The sad thing about this system is that its basically really difficult to change, even assuming technological change. Ninja's and Nobility have a good thing going on and have all the power, and little incentive to change the system. The Serfs who do the actual work and would love to change things dont have to power to do so.
In real life, feudalism transformed into our capitalist society for various historical reasons, but I see little reason why it should here. (I'm gonna simplify things a bit here) One of the necessary factors was that in western europe, serfs essentially gained more power and became free and more powerful workers because of the great changes that occured thanks to Black Plague. However, in eastern europe, the same Black Plague happened and overall, the landlords won, and repression grew worse.
In a world with ninja its just plain hard to imagine the commoners gaining at the expense of the nobility.
Sidenote: I doubt that peasants in the Elemental Nations have the same "privileges" a serf has, and suspect that in general most are essentially slaves. Though its up to Sixperfections exactly what the details and institutions of the elemental actually are like.
Edit: this is sorta mostly complete speculation by me. It doesn't quite fit with canon naruto, which makes sense because kishimoto is a mangaka not a historian. It's mostly based off of what I'd imagine the society to actually look like if you apply a bit of realism.
Canon naruto is sorta a confusing mess.