It's not very close. The South Entrance to Yellowstone may not be that far but that is all there is, an entrance. To get anywhere in the park (though the drive along the Lewis River Gorge is spectacular) is 25 miles to Grant Village. The majority of the thermal features are even further. I don't think there would be a problem with leaving an empty tent - there are usually other campers around at each of the Grassy Lake Road primitive sites. There are bear boxes for food and other items - just be bear-aware!

About 2 miles from the south entrance. There are grizzlies in the area, so that’s up to you on how comfortable you feel in a tent. Hope this helps :)