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I don't have an answer for how to remove the smell. However, your landlord should remove the mothballs because the fumes are toxic, according to the EPA:
http://www.epa.gov/ttn/atw/hlthef/naphthal.html

I have a similar problem. The house I lived in during my last 2 years of college had closets that reeked of mothballs, so now I STILL have clothes (almost 3 years later) that smell like them. Any magic anti-mothball solution out there?

Get a box of baking soda, preferable one of those ones that's made for the fridge and has the vented side.

Stash it as close to the smell source as possible and change it every 3 months or so.

Use blocks of aromatic cedar instead. It's just red cedarwood. It smells phenomenal, is completely natural, and is a natural repellent for moths.

"WARNING: Never EVER allow your children to play near a mothball odor filled area. This chemical is not just toxic and carcinogenic, it acts as a neural destabilizer that can retard intellectual growth over time. And it doesn't take much, just long term exposure."

Tell the landlord to remove them or else. Seriously.

An ionizer along with washing everything (including floor, ceiling, and walls) with oxyclean will help get rid of it.

I hate to say it, but unless this is New York City...you should move.

Have a talk with your landlord and tell her your concerns about the odor. Ask her to remove the mothballs and use cedar chips or something non-toxic.

I second everything Jason said.

smelleze mothball deodorizer help too :http://www.imtek.biz/page/N/CTGY/sp-moth

Not to alarm you, but my great uncle was obsessed with keeping moths out of his stuff and his whole house reeked of mothballs, as did any Christmas or birthday gift he ever gave me. He died of cancer and we're all pretty sure it was a result of breathing in toxic mothball fumes all those years.

Print out the stuff from the EPA, take it to your landlord and demand that she remove them immediately. In the meantime, I wouldn't stay there.

current health news: mothballs are def. hazardous!!!!! Poison/do not breathe/etc. etc..redmakesmehappy and dearmisha are absolutely correct!!!!!! This is a well known carcinogenic !!

if you want even more ammo for your landlord:

http://en.wikipedia.org/wiki/Mothball

Naphtha is TERRIBLE for you and your pets. Get the landlord to remove them and then get yourself an air purifier.

Hmmm...storage space...she's not trying to cover up the smell of decomposing flesh in there, is she?

thank you for your helpful suggestions! to answer some of your concerns: my landlady claims she has removed all of the mothballs, so I think I'm kind of SOL there.
Any suggestions for removing the smell or perhaps deadening the negative effects of the mothballs?

Charcoal and coffee beans also absorb odors. Use real charcoal-lump wood, not pressed briquettes. To counter the odor, try some safe essential oils away from the cats. Lavender or mint oil drops on top of broken balsa wood in a glass jar with a few potpourri flowers should counter the smell. Place the jar away from the cats or cover it with a square of cute fabric and a rubber band. I own a rental property and these tricks have worked to get rid of tenant odors. The charcoal and coffee are great for the fridge too.