Yes it is possible for an advanced country to have a monarchy.
Yes it is possible for an advanced country to have a federal political system that is partially feudalistic, with hereditary lords and vassals.
No, it is not possible for an advanced country to have serfdom, so if you are thinking about serfdom as a vital feature of feudal society you are out of luck.
Today there are about 180 to 200 countries on planet Earth, depending on which polities count as countries in one's definition.
And some of them are more advanced than others, but in every one of them at least a proportion of the population has a degree of access to the most advanced products developed in the more advanced countries. Almost every country in the world is part of the world wide technological civilization.
So advanced countries in the world today have various forms of government.
Most countries have republican forms of government.
There are parliamentary republics where the prime minister is head of government and the president is a ceremonial head of state.
There are presidential republics where the president is both head of state and head of government.
There are semi presidential republics where the president is head of state and the executive powers are shared by the prime minister and the president, making them both partial heads of government.
And today most republics claim to be democratic. Republics generally have positions on the democratic scale between very democratic, mostly democratic, largely democratic, slightly democratic, and not democratic at all, since many republics have authoritarian, dictatorial, or totalitarian governments.
A number of states in the world today are monarchies. Monarchies generally share the same sort of distinctions as republics.
Some monarchies are like parliamentary republics, having a prime minister as head of government and a monarch as ceremonial head of state.
Some monarchies are like presidential republics, having a monarch as both head of state and head of government.
Some monarchies are like semi presidential republics, having a monarch as head of state, and with both the prime minister and the monarch having some of the powers of head of government. In most monarchies today the monarch has at least a tiny little bit of executive power, while the prime minister usually has much more executive power than the monarch.
And today most monarchies claim to be democratic. Monarchies generally have positions on the democratic scale between very democratic, mostly democratic, largely democratic, slightly democratic, and not democratic at all, since many monarchies have authoritarian, dictatorial, or totalitarian governments.
Every Country in the world today, except for tiny little Vatican City, has administrative subdivisions, whose governments are either elected by their residents or appointed by higher authorities.
Some countries, like Azerbaijan, The Bahamas, & Bahrain, for example, have only 1 level of subdivision. The United States of America, has three main levels: states, counties, and municipalities. The People's Republic of China, France, Myanmar, and Pakistan have five levels of administrative subdivisions.
And of course those levels of subdivisions are interesting because in various countries the relations between the levels resembles a feudal monarchy more or less - usually less.
Federalism is the mixed or compound mode of government, combining a general government (the central or "federal" government) with regional governments (provincial, state, cantonal, territorial or other sub-unit governments) in a single political system. Its distinctive feature, exemplified in the founding example of modern federalism by the United States under the Constitution of 1787, is a relationship of parity between the two levels of government established.1 Federalism can thus be defined as a form of government in which there is a division of powers between two levels of government of equal status.2
Federalism differs from confederalism, in which the general level of government is subordinate to the regional level, and from devolution within a unitary state, in which the regional level of government is subordinate to the general level. It represents the central form in the pathway of regional integration or separation, bounded on the less integrated side by confederalism and on the more integrated side by devolution within a unitary state.
Leading examples of the federation or federal state include the United States, India, Brazil, Mexico, Russia, Germany, Canada, Switzerland, Argentina, and Australia. Some also today characterize the European Union as the pioneering example of federalism in a multi-state setting, in a concept termed the federal union of states.
In a typical medieval country there would be both feudalism and the manorial system. The holder of a manor would be the lord of that manor.
Counts were originally appointed officials but made their positions hereditary. In the Feudal system a count would be the overlord of most of the lords in his county.
Dukes were originally appointed officials but made their positions hereditary. In the feudal system a duke would be the overlord of most of the counts in his country.
And of course a king would be the immediate or ultimate overlord of all the lords, counts, and dukes in his kingdom.
So in this greatly simplified model of the feudal system there would be four levels of feudal rulers: lords, counts, dukes, and kings. And each feudal ruler would be the vassal of some superior ruler or rulers for his fief or fiefs. And vassals and overlords had mutual obligations and rights.
So there are some similarities and differences between Federalism and Feudalism.
It may be noted that one federal state, the German Reich from 1871-1918, had most of the member states ruled by hereditary monarchs (with more or less democratic governments) with feudal titles, thus having considerable overlap between federalism and feudalism.
At the present time the United Arab Emirates is a federation of seven emirates ruled by hereditary sheikhs, with the president and prime minister elected by the sheikhs. Thus the president is both a sheikh and a president, sort of like the president of France is a co prince of Andorra.
Malaysia is a federation of 13 states, nine of which are hereditary monarchies. The nine monarchs elect one of their number to be the head of state, the Yang di-Pertum Agong, every five years.
So the answer is yes.
Yes it is possible for a country at early 21st century levels of development to have a monarchy, since some of them do.
Yes, it is possible for an advanced country to have a more or less feudalistic type of government, like a form of federalism with hereditary rulers.
But no, it should not be possible for an advanced country to have serfdom. All forms of serfdom or slavery tend to slow down progress and development. So in an advanced country slavery and serfdom would be illegal, and the less widespread any similar and possibly illegal status is, the move advanced the society will be.