I found an old clam-lid tote tray that I store my two grease guns and tubs of bearing grease in. I took an old piece of grill or oven rack, trimmed it to fit, set it on a couple of 2x4 pieces of wood in the bottom of the bin, and put a layer of Absorb-All in the bottom of the bin. The grease guns sit on top of the rack and can drip into the Absorb-All.
For me, it wasn't so much the grease dripping out the ends, as the weeping and migrating of the oil in the guns and the tub of grease. It seems they always end up getting covered with a thin layer of grease on every external surface, which ends up migrating to the shelves and anything else that might happen to touch them, but it also collects dust. They're just dirty tools, and there's no way around it.
Closed in the bin, they sit on a shelf easily enough, the bin stays clean, and everything else in the shop is protected from getting greezy.
Cost me nothing... Found the bin on the road, recycled the rack from a piece of grating I picked up somewhere.
-Brad