If we did say have full scale nuclear war then it is quite possible to blow enough dust into the atmosphere to actual cause problems with food production. That's the real danger, no literal winter. Even a relatively small change in the amount of sunlight that reaches the surface of the Earth could cause mass famine, even if the world were more or less the same temperature.
Fallout in and of itself is an immediate and short term threat, unless you have a lot of irradiated dust that travels in the wind. The big problem with fallout is ultimately that irradiated dust can get into food supplies, or travel huge distances if the dust is launched high into the atmosphere.
5000 years would be enough that radiation short of leaking sci-fi fusion reactor would clear of radiation. Despite what Fallout would suggest radiation does not work that way. Now, that being said if you had a direct nuclear ground strike it is possible to produce an area where nothing will grow, although that area won't be very big and after 5000 years is probably going to be covered in a thick layer of top soil.
In many ways if you wanted to move to the age of sail 5000 years after some disaster you'd need to push humans back to only knowing what they knew at the time Stone Henge was being built, so some 3000 BCE. We're talking early Bronze Age here, if not before depending on region. This would literally require bombing each other back to the Stone Age.
The only thing that I can directly think of that would preclude an advancement in learning, and eventually a regression in technological levels would be loosing the ability to read and write. Those two things are key to teaching and learning skills without having somebody on hand to directly explain what they already know.