Doing the "shill job" quest for the thieves' guild, and being asked to plant a stolen ring. I am in a backroom of a house, staring at the quest marker right in front of me. But, I have no idea how to plant the item.

Open the chest they want you to plant it in, find the ring in your inventory, then put it in the chest. At least one of the times I got this quest, Vex gave me the wrong item to put in the chest, and so the quest was bugged and uncompletable. If so just go back and tell her you can't do it.

There is no chest, but a bunch of crates and barrels.

Whichever storage place the arrow is over is the right one.