I woud vent out of one side of the closet and then rely on passive intake for air coming in from the other side. "Fresh" air is a bit a myth. There is plenty in a home.
i agree, my grow box is in my bedroom, and some days it only gets the air that is available in that room, ie windows are closed door is closed, the plants dont have lungs like us humans, and we would be dead in minutes in a grow box,, but not in a bedroom, and besides they give off what we need, and we give off what they need ,, so its a good realationship,, that being said,,
yes you need a good carbon filter, I am using a 6" that i bought on ebay, its in my second crop in flower,, and still no smell at all in my bedroom,, your going to want to get maybe a 4 inch inline fan, and a speed controller, and a 4 :" carbon filter, since its in a closet you can put that on the outside of your grow box if you want, try to position the vent hose to it points out of your closet some how, thru a louver or leaving the door cracked, try to make your intake be on th eother side of th ebox