Legally you can drive on whatever road you'd like in whatever vehicle you'd like, but coverage in case something goes wrong is probably not included in your rental company's coverage. The "4x4 only" signs you referred to are meant to discourage tourists who don't know any better.
Depending on the road, weather, your driving skills etc. you'll easily be able to traverse certain "4x4 only" roads even in a VW Golf. The trick is knowing which roads those are, and accounting for the tiny stream you passed on the way in potentially having turned into a river due to rains or snowmelt since you drove in.
