18-20 miles the short way over Deer Creek Pass. You can get into the area multiple ways, from inside Yellowstone, up a couple different creeks on the North Fork before you get to the east entrance of Yellowstone. These routes I believe are a little longer but not quite as steep or rugged.
The trail in from the Nine Mile trailhead inside the park isn't bad, but it's a long way.
There is a trail that goes from the Elks Fork south over the top and into the head of open creek. There really isn't a pass there though, you are pretty much right on the peak. This way would be pretty high and then you'd have a pretty good trip down Open creek to the Thorofare.
The trail in from Moran at Turpin meadows would be pretty easy but again long. It takes about 8.5 hours horse back, with only one pretty easy pass. This is the way we usually go but we hit our camp before the Yellowstone or the Thorofare.
Like Partagas said there are several ways in. But none of them are short or easy. I see a few backpackers in there every summer. Usually they are resting in the middle of trail to scare my mules.
