For a place to pitch your tent or unpack your picnic, find yourself at Cow Creek Campground.
It's got only a few primitive campsites, and they're first-come, first-served, so your best bet for snagging one is during weekdays or non-holiday weekends. Ain't no potable water, ain't no trash pickup, and ain't no firewood available, so BYO both in and out, enjoying the crap outta nature in-between.