I may have been exaggerating slightly when I say "there is no place," for there probably is. Still, the East is what my father would have called "thickly settled," at least in most places. But get a little further than my completely arbitrary 100 mile radius and there are places where you could easily get lost (or stay hid) for longer than you'd want (or for as long as you want). But the dangers are also easily exaggerated, though hardly non-existant. Personal carelessness is probably the greatest danger, from falls and whatnot. If you are thinking of a place where you're likely not to run into another soul, then the danger from other people would have to be close to zero. Animal dangers vary widely of course but I always suspect claims about packs of wild dogs.
There are places where private planes have gone down and not been found for a few days but unless you're flying, you probably wouldn't be there either.
By the way, I consider twenty miles to be a nice day hike for about a ten hour day if there's no snow on the ground. If you had a busted knee or a turned ankle, it would no longer be a nice day.
Shoot low, sheriff. They're riding Shetlands!
Underneath the starry flag, civilize 'em with a Krag,
and return us to our own beloved homes!
Buy War Bonds.