Wait, do you want something better than Mad Max, or do you want something as good?
Actually, I merely want a realistic timeframe (the world as presented in Fallout 4 would make a lot more sense if it was 5-20 years after the big bang), along with a visible effort at reconstruction.
If you'd actually extrapolate a 300 year timespan, the world would be utterly unrecognizable and alien to the Sole Survivor. Take the Black Death that wiped out large swathes of Europe in 1347, a period which was technically rather stagnant. Now look at Europe in 1680. The plague had profound religious, social and economic impact, sure, but you'd be hard-pressed to see any trace of it in the landscape.
Now consider the 1950s retrofutur setting, where technological advanced were made by leaps and bounds, and at the same time relying on comparatively fragile technology and a very finite power source - reactor grade uranium doesn't exactly grow on trees, you see. At that point, you could expect anything from "Crystal Spires and Togas" to something along the lines of "Warring City States in the Wastelands". Seeing that the chinese supposedly won, presumably with a heavy touch of either communist or imperial china to boot. It'd be bizarre at best, with the distant past being some obscure side note.
Hell, 300 years in that setting is probably enough for 2-3 extra apocalypses. And don't get me started on the fact that 300 year-old junk is still supposedly working, with just a few dinges and paint scrapes. Not to mention those incredible amounts of powder-based ammunition that still fires after sitting on the shelf for several centuries. Or those fusion cells that manage to power a factory for 300 years, but crap out after 15 minutes of being put in a suit of power armor. The whole world makes so little sense that it's not even funny anymore.
Check Fallout 2, which does a much better job at worldbuilding, and going along with the inherent campiness of the setting.