When your chest of drawers has odors in it, your clothes and other items will pick up the odors if they aren't causing the bad smells. The first thing to do is empty the drawers and wash the contents. Then if you find the smells remain--maybe from mothballs, musty or new wood, laundry soaps, drawer sachets or spilled perfumes--work on the drawers to eliminate the odors. If the odors penetrate wood in a chest of drawers, they will be more challenging to eliminate.
Use white bread and white vinegar. They will neutralize odors from your chest of drawers. Place one slice of white bread into a small deep bowl and saturate the bread with white vinegar. Place the bowl inside the smelly drawer and let it sit for 24 hours. Use a bowl that is deep enough so the vinegar will not spill when you open the drawer.
Try newspaper. Wad up newspaper to fill the entire space of the drawer. The newspaper will absorb the odors and get rid of them. Replace the newspaper every three days. This may take up to two weeks to be effective.
Use baking soda. Pour a box of baking soda into a bowl. The baking soda will neutralize the odor and freshen your drawer.
Try wood shavings. Cedar or pine shavings may be effective in getting rid of smells from your drawers. Place the wood chips into bowls and put them in the drawers. The wood shavings will reduce the odor and absorb any moisture. Moisture frequently causes odors.
Use charcoal. Charcoal effectively absorbs odors. Place a few charcoal pieces into small bowls and put them in the drawers. Let the charcoal sit for 24 hours and then check to see if the odors are gone. If not, let the charcoal remain for another day.
Use elbow grease. In a bucket, mix mild dish-washing soap with hot water. Use a clean cloth and submerge it in the water and wring out the excess water. Scrub the insides of your drawers, making sure you clean all the surfaces. This may wash away odor-causing residues that were on the wood. Make sure your drawers are dry before refilling them.