Just about any mature pre-industrial agrarian society will have some significant features of feudalism.
A divide will tend to emerge between a landed aristocracy and an unlanded peasantry. Malthusian effects make yeomanry unstable in the long term once all available arable land has been concentrated: successful yeomen will tend to accumulate wealth and land, while unsuccessful yeomen are likely to be forced to sell, and any families that have multiple surviving heirs will have to subdivide their holdings (leading to penny-parcels of land too small to support a yeoman family) or concentrate inheritance on a single heir (leading to the main line of heirs holding all the land, while secondary branches are left landless).
A military caste will tend to emerge, and they'll tend to either be the landed aristocrats or be loyal clients of the landed aristocrats. Being an effective warrior with pre-gunpowder weapons is a full-time job that takes a decade or more to learn well, and good weapons, armor, and warhorses are horrendously expensive in a pre-industrial society. The only people with enough money to support these full-time soldiers will be large landowners. Peasant levies and yeoman militias did exist, but they were generally used to bulk out an army formed around a core of knights and men-at-arms, and were rarely effective without that core.
This military caste, or whoever commands their loyalty, will tend to dominate government, simply because government is based on the organized use of force.
A decentralized power structure based on personal loyalty and family connections will tend to emerge. As is frequently observed on this board, the fatal flaw of the Roman Empire was size vs. travel and communications technology: if it takes a month for the Emperor to hear what's going on in a frontier province, and then another month for him to do anything about it, then his ability to rule is very flawed. His best bet is to put someone he trusts in charge of day-to-day and even month-to-month affairs, and give that someone enough direct power to manage most problems on his own. And without the kind of information and legal institutions seen in modern first-world countries, that level of trust almost always has to be based on personal loyalty and family connections.
That said, though, there's lots of ways that you can mix things up. These are only tendencies, not laws of physics, and deviating from the feudal model somewhat can be made to work at a moderate stability cost.
One alternate model often seen historically was for a strong central government to exist based on a revenue stream of plunder and tribute from neighboring states or conquered client states. This revenue stream can then be used to train and pay long-enlistment professional soldiers who do the job of a military caste. Regional governors would still be needed, and would still be tied to the central government by personal loyalty, but the central government with its professional army would be much better able to make and break regional governors, making them appointed officials rather than feudal magnates. This was more-or-less the model the Roman Empire followed, and many Middle Eastern states in classical times followed a similar model.
Another alternate model is the Mandarin model seen in China, where a non-hereditary class of scholar-bureaucrats, centrally educated and loyal to the system that gives them status, wield enough governing power on behalf of the central authority in order to significantly dilute the power and status of the military caste and the landed aristocracy.
It'd be very interesting to come up with a way to make the Mandarin model work in Medieval Europe. There were already some seeds of the model present in the early Medieval period, with the monastic education system creating a class of scholars who collectively had a near-monopoly on literacy and non-vocational knowledge, celibacy rules to prevent church officials from turning into landed aristocrats, and a formal church hierarchy with the Pope as central authority.
