This is really not a 4x4, 2wd Hc will do fine unless wet and then you should stay off most of these roads with clay stretches anyway.
Its about 13 miles one way to the Limestone Crossing in Johns Canyon. The road is very good, snakes along at the base of cliffs offering good views into the canyons cutting to the San Juan and over the top to Monument Valley. At the scenic Limestone Crossing, with it's broad sheets of bluish limestone gleaming with water and cottonwood trees this would make a good base camp. Cows do graze in the area. Along the way on the scattered varnished boulders petroglyphs can be found. If you do have a HC vehicle it is possible to cross at Limestone Crossing carefully, and drive up or down John's on a low grade track. Toward the San Juan it has a very bad spot about one more mile on, park unless you have a quad or dirt bike or minimum 33 inch tires, lockers, a winch and narrow wheel base.
More petroglyphs can be found but I was not so ambulatory at this time. You can walk on the old mining road all the way to the Slickhorn confluence and beyond.
I also car camped above overlooking valley of the gods and will include some pics of that area.