My son and I have been twice. The first trip was marred by car issues, but the second trip, we were able to actually camp in the Big South Fork National River and Recreation Area.
We parked at the Twin Arches Loop National Recreation Trail head, then made our way down into the valley to view the two gorgeous and huge arches there.
As the sun was setting, we found a good place to set up our hammock gear a little ways off of the trail, then we fixed our supper at a nearby over hang, and finally snuggled in for a wonderfully cold night (late march). There was ice on the trees in the shade of some of the overhangs.
It is a nice hike, plenty of interesting areas to set up camp. Fair warning, some of the "stairs" down into the canyon are steep so be prepared.
Make sure you have your permit with you while camping and have a wonderful time in this beautiful valley.