The roads are paved to all of the major beaches (you're not allowed to drive on the beaches). Several of the beaches require a hike once you have parked nearby (Honeymoon, Francis, Waterlemon). The Lameshur beaches require 4WD but the others are accessible by standard sedan.
The roads on STJ are very steep so a 4WD provides a 'comfort zone' for some. They are especially helpful when the roads are wet and become slippery but not absolute necessity to get around.