I suggest the following ingredients to make Feudalism more believable:
* a very depopulated world
* loss of agricultural productivity.
Loss of agricultural productivity
You need loss of agricultural productivity and a tie-in to land for agricultural produce. More on that later.
You can have a loss of agricultural productivity by climate reasons. Or by loss of modern technologies, including any and all that make food supply ample. But low agricultural output and low productivity in general, even combined, do not require the foregoing of all post-industrial technologies and know-hows.
For example, say 90% of the remaining population on earth is required to produce sufficient food in aggregate. But a highly automated factory complex produce radios with few human input -- sure you always have upstream input but you can certainly have designs that suit simplified inputs. Said factory may be able to produce enough radios for 5% of the earth's population annually -- allowing some to always have new ones and the rest to only work with old radio sets that may become faulty in some way due to age. In this scenario, you could say a lot of technological advancement is included in the production of radio, so much so that very little input is enough to produce for the whole world. Yet overall productivity of the world is low because food is required and food is being a bottleneck for greater economical development.
Loss of agricultural productivity can be important for Feudalism. Feudalism was a land-based social order where
1) workers were tied to the same parcels of land all-year-long
2) nobles had the authority and ability to tax production from (larger) land parcels
3) a hierarchy existed to combine tax over large geographical areas and thus support things such as a military that can provide stability and order and some luxury items that helped people feel that the authority was deserved and proper.
If people can move around freely and still be a part of a productive process, then they won't be willing to stay under some abstract contract that bind them to some designated noble -- however relatively un-educated they may be.
So for the sake of Feudalism, you should tie 90%+ of earth's remaining population to daily working over the same pieces of land.
Note that low variability in agricultural productivity is also needed. ie. You shouldn't have a single place that is highly productive and nowhere else that can produce food. If the only place that can produce starch based food is in the UN headquarter building in a city named New York. You can have whatever social order that makes sense at UN. 222
Outside of it, people are not tied with land anymore.
That said, a generally unproductive world but a few pockets of productivity -- even within agriculture -- can make for interesting dynamics. Like 0 everywhere vs 100 at one place wouldn't work. 1 vs 2 on the other hand could be interesting.
A depopulated world
A depopulated world does a number of things in Feudalism's favor, including:
* difficulty to maintain law and order over vast areas (and the consummate incentive to banditry)
* difficulty to maintain education
* even more difficulty with aggregate output that feeds into everything above
* the same sequence of events could have destroyed the social/national institutions in our real world in the same stroke
The implications are self-evident. Without the large population the modern earth has and supports, the historical benefits Feudalism was supposed to provide are not easily replaced via alternatives. They included social cohesion and stability of life. The key is to take away alternatives.
What loss of agricultural productivity achieves is that people are unable to move around and be productive other than being peasants of their specific pieces of land. Depopulation takes away plausible alternatives by not permitting individuals with free time. For 90%+ of population, you work your field for 8 hours a day or your starve. Otherwise, when people have free time, they will try to do things that break the current order. Say there is no difference between a republic where a government imposes tax and a monarchy where nobles do in term's of people's daily experience. If you give people free time, given the communication technology, they may still rebel against a Feudal system and put a republic in place out of preference. That's why free time should also be difficult to come by.
Perhaps you can also introduce other factors that accentuates the lack of free time. Perhaps, to make agriculture work in that post-apocalyptic world, farming is very complex -- you need recurring efforts to deal with certain side-effects from radiation, maybe add some new micro-organism that people have to work against -- while being unproductive relative to our real world.
And there you go, I think with some details, Feudalism can become realistic enough.
Yet at the same time, you are not restricted in having modern transportation, modern communications, modern entertainment, etc. 222