Interestingly enough, the Fukushima exclusion zone has roads that pretty much fit the bill. They're the pink lines on this map (PDF), with the exclusion zone in gray:
https://www.meti.go.jp/english/earthquake/nuclear/roadmap/pd...
You are permitted to drive through but not stop or get out of your car. I presume ordinary speed limits apply though, so no exciting post-apocalyptic signage.