A lot of answers here talk about how innovation is intrinsic in humans (and hypothetically other intelligent beings) and thus so is progress. But no-one seems to have mentioned the key underlying part of human progress which is sharing and co-operation. It doesn't matter if you have a genius like Leonardo Da Vinci if everything he knows dies with him. Humans have only progressed so far because we share our knowledge with each other, spread it around and keep having more people improve on it, then share those too.
The things that make this sharing possible are of course language, but especially written language. Medieval periods are obviously times when very few people know how to read or write. You could experiment with there being no written language at all, or it being very clunky and awkward, or learning to read being rare or hard.
But beyond that, the other key impediment to the sharing of knowledge is education. elemtilas touched on this with their suggestion that your people should be
clannish and traditionbound in the sense that when the elders teach
the younglings, the younglings accept without question and repeat the
same patterns over and over again over countless generations.
How about a better idea? Education is limited to your genetic heirs only.
Someone else mentioned indigenous Australians and other so-called stagnant cultures, and suggested they were stagnant because their needs were met. Not so. Most of the stagnant cultures on earth are in places where survival is extremely difficult. In the case of Australia, the environment is not only so naturally scarce of food that you can't have high population levels, but without frozen winters a key method of food preservation used elsewhere in the world was impossible. Possibly as a result, a culture developed whereby the location and understanding of how to find food/water in an area was secret knowledge.
The ancient Australians didn't have a concept of individual land ownership because they didn't need one - to learn to survive in a new place was extremely unlikely, especially if people already lived there. And if the people who lived there didn't share that knowledge, there was too much risk and not enough benefit trying to do so yourself. People raided each other for resources and to hunt large animals when they were present, people traded, and people were given food when they passed through each others lands, but that was the extent of resource sharing between communities. This seemed to extend to the few places where some level of "farming" or tending existing grains was possible (though the nutritional yield on these is probably paltry compared to even ancient wheat and grains). People just didn't share with outsiders the knowledge of how they survived, and so movement was limited.
Another aspect where this ties into medieval societies is the use of trade guilds. They closely guarded their techniques in a similar way to the Australian cultures, and required membership to gain access.
So theoretically, progress would be severely stunted by a society-wide adherence to only teaching your own children what you know. Combined with a lack of writing, this stops the sharing of knowledge to barely perceptible leaks. Only extremely bright people who actively want to learn something else and have good observation and complex reasoning skills could learn things at a usable level. And if professions were inherited like the knowledge was, even if you learned new things it would be hard to put them into practice. These bright people would have to move to new areas they weren't known to pose as someone who should have that knowledge.
You could either create a quasi-guild system whereby in all places people of the same professions have secret signs to identify each other, or you could create reasons people simply did not move about, as in Australia. Aboriginal knowledge was not limited to genetic children, but it was culturally reinforced quite heavily. This would also be an important factor.
This could solve the "how did they get to this stage" problem. Either they haven't been at this exact level of progress forever but have just been progressing at such a glacially slow pace it seems like it from an individual perspective. OR, they didn't always have a system of guarded or secret knowledge.
You could have some social ramifications from this kind of thing, such as some professions 'ransoming' society to keep producing, but this would require collusion of all their supporting professions that supplied materials. Workers from those professions could always be poached from other communities in that event though. Of course, even other people of the same profession wouldn't know exactly what their fellows knew, because they weren't sharing knowledge between each other like a proper trade guild. So even if a specific individual family develops some innovations, those techniques are only ever known by their heirs. A couple clever generations can be diluted by some dimmer generations who perhaps don't quite "get" what granddad was talking about.
If there was conflict between communities, targeting all the members of a vital profession could be a valid military strategy - another hiccup for the idea, because an entire profession almost dying out would be a possible impetus for breaking the system.
The flow-on to gender roles could also be interesting. My first thought is women would have to taught as well as men, since not everyone is guaranteed sons, and then work at least until they were married. But, to avoid inbreeding, women would have to marry outside immediate family. That might require vast cousin systems for inter-marrying, thus creating "tribe guilds" of professions, and that still creates the chance for knowledge to spread inside that profession, which you'd want to avoid. If women married men of different professions, this creates the opportunity for profession mixing. So in that scenario perhaps women were not taught at all, making their position similar to medieval Europe or even more restricted.
Alternatively, since the whole "not guaranteed sons" thing is still in play, parents could nominate offspring to be their heirs, either male or female, and once they had enough of these (for whatever enough means) any further offspring were not taught. These younger children would be relegated to extremely unskilled labour (think ditch-digging). An heir child could marry a non-heir child without fear of blending knowledge. Though this creates a system of elderborn and youngerborn children almost being castes. (Probably better than "ditchdigger" being an inherited caste like "untouchables" though.) If families help out all their members regardless of task that absorbs any resentment. You might need a bit more society-wide sharing to offset economic needs that would make people break the conventions.
A third option is that there aren't traditional marriages; instead women simply get pregnant by men as they like, then those children are raised in their maternal home. You might develop a slightly matriarchal society like this though, which would look different to medieval Europe. If your males are anything like human males in terms of their sex drive and being extremely competitive for mates, you'd need all the ladies to be pretty sexually active across the board to avoid conflict, and this creates biiig problems for disease in medieval tech levels. So unless STDs are not a thing or you work the high death rates into the social system some how, that might not work out.
There's probably lots more little wrinkles I haven't identified, but the crux of the idea is that when people don't share their knowledge, progress is so limited, so slow, so stunted and easily reversed, that it would barely exist at all. If a culture believes this is the best system (for some reason you would have to invent) and there's no written language for people to even share philosophical and cultural ideas, it would seem to create quite the technological stalemate.
