Try spraying plain vinegar around the area.

White vinegar is also a natural cleaner/deodorizer. You could try making a paste with baking soda & vinegar and cleaning the area, as well as the trash cans themselves. Then, rinse the cans with more white vinegar and let dry. If your cans are plastic and very old, they may just be beyond saving, since some plastics seem to really harbor smells, and you may have to bite the bullet and buy new.

Scrub the area down with TSP (use gloves and goggles) then paint it.

It depends on the area and what the smell is. If you're taking about a cupboard area, bring it to the attention of the landlord. It's possible that there are rotting wood board or floorboards somewhere and it's their job to replace it. It actually works out in their favor, since that kind of thing can get them dinged on an inspection.

Vinegar, lemon juice, tea tree oil and lavender oil are supposed to be antiseptic, antifungal, and smell nice. Charcoal and/or volcanic rock is supposed to absorb odors too. If the smell has soaked into any wood or plastic surrounding your garbage area, you're going to be in trouble.

You could try using a pet-odor product with natural enzymes to eat/destroy smell-creating bacteria. Not sure what it would do to non-urine smells, but I used a bunch of it on a non-functioning bathroom in a historic house that a group of people who rented the place for a party (with lots of alcohol) thought was a functioning bathroom. With two big boxes of baking soda, the pet odor wash got rid of the smell that lingered even after the place was thoroughly cleaned. Don't forget to wash baseboards and walls in addition to floors.

The Bad Air Sponge is kind of awesome. I use it to keep the kitty litter smell from permeating my apartment (I use a natural, wheat based kitty litter which is great for my Gato and the environment, but not great at smell mediation).

When I had to de-stink-ify the refrigerator in our new home (which hadn't been used for months prior to our moving in), and no amount of vinegar, baking soda, or chemical cleaners seemed to be making any difference. I followed the suggestion of someone, somewhere on the Internet and used pure cinnamon oil. I poured a small amount into a shallow dish and left it in there for a few days. Then I wiped the surfaces down with a bit of the oil on a sponge. At first I thought it was only partially successful, but after we moved in and started using the fridge, the smell was completely gone.

I actually read that baking soda absorbing smells is myth. What I found worked great in my last stinky apartment freezer was a tray full of charcoal. I just used charcoal briquettes and the smell was gone in a few days...don't know if this would help in the garbage area...but couldn't hurt.

I've had good luck with borax and hot water-- and then you can dump some more in the bottom of the pail after it's clean and dry too.

Paint it.

agree with Rhoswen, you need to investigate and figure out why it smells. If the floorboards are rotten, or there's a dead animal or water damage, no amount of vinegar will fix it.

I agree it may need to be revamped, sealed or painted. The landlord should have taken care of this before you moved in.

Without a doubt, the holy grail of enzymatic cleaners, AIP (stands for Anti-Icky-Poo). Silly names aside, this stuff is amazing. It's sold as a stronger grade pet accident cleaner, but it works beautifully on any odors caused by biological material. Nasty smells in the trash area are just that, and AIP will take care of them. It takes a few days to a week for the enzymes to chew up the bio stuff causing the odor, but it definitely works.

This recommendation is based on experience. We bought a home owned by cat hoarders (unbeknownst to us prior to closing), and there are no words to describe what we found in the "new" carpet and drywall. There is nothing quite like the pungent, impossible stench of cat urine. After unsuccessfully trying to cancel the sale, we were stuck and desperate. We yanked out all the flooring and mopped the concrete and drywall with AIP. After less than a week, the odor was completely gone and all stains gone under UV. We have used it since on trash areas, garbage cans, and even the garbage disposal with brilliant results. Silly name, but it works.

These are great suggestions. I hope you figure it out, and when you do, I would suggest keeping a small bag of food-waste/smell-prone trash in the freezer instead of throwing it in the trash can to take out in a few days. We put all of our compostables in the freezer, empty when it gets full every couple of days, and that way our garbage area never smells. I also keep baking soda sprinkled in the trash bin, though now I question since a commenter said that baking soda smell absorbtion is a myth. If you have a hard time filling your freezer (which we absolutely do NOT) it also helps it run more efficiently, so they say.

@RiverJai That is an awesome recommendation! Terrible story, though: so glad you were able to fix it!