I moved into a very old Victorian building in San Francisco last June.
The wood floors were re-done and the place was re-painted. Now that
the new paint smell is gone, the old apartment smell has really set
in. It smells like a grandma's house-very musty and gross. I didn't
realize how bad it was until I went on vacation and opened my suitcase
and all my clothes smelled musty and like a grandma. When I got back
to my apartment, I noticed the smell even more. It's the worst in one
of the coat closets. Living in San Francisco means I wear a coat
almost every day of the year and I hate that my coats smell so bad!!
Since I live in a small studio, I need the closet space, so I can't
just move them to a different location.

Are there any solutions to this? 

All of the solutions I have thought of will not work:

For example, the rest of my clothes are in an open area, hanging up
and folded on shelves in a small nook closet, they can't be tucked
away into a good smelling dresser because there is just no room for
one. I don't think Fabreese-ing them will help, because they will have
to be hung up in the nook that smells like my apartment, so the smell
will just get on them again.  I also think scented candles might help
mask the problem, but not fix it and also they wouldn't fix the
clothes smelling bad problem. I can't open windows because it's so
cold outside!!

What do I do?? I can't smell like a musty 90 year old anymore! Any
help would be much appreciated! Thanks so much!!

My short-term recommendation is to buy some cedar wood and put it in
your clothes closet(s). It does not get to the root of the problem (an
expert Answerer will need to address that) but it is a fast,
pleasant-smelling way to start. If the cedar smell begins to fade, a
quick hit with some sandpaper will bring it back.

Cedar is available as shavings and small blocks (some have a little
hangar attached), but I'd get a few planks and put them up.

I wonder if you might have mildew or mold in the apartment caused by
moisture.  That can really give you that lovely "basement" musty smell
;).  Can you pinpoint where the smell is coming from?  I would check
to make sure you don't have mildew or mold growing between the walls.
Or, if you have a leaky pipe that's just lightly dampening the wall
under a sink/toilet, that can emit an odor. We get that lovely smell
in August in our basement and we turn on a humidifier when it gets
bad. A plumber can also be called to check for moisture if you think
that might be the problem.  They have a meter that they use to check
for it.

Good luck!

Three possibilities to consider:

1) Some people have had success with an open container with 1/2 to 1
cup of white vinegar in opposite corners of the room (refill container
as needed.)

2) Several pieces of charcoal briquets (in a large coffee can with
holes drilled to assist airflow for example) hidden may also help.

3) The more expensive and possibly most effective solution: purchasing
an electrostatic air purifier - as sold by Sharper Image or Oreck. 
Either will run in the several hundred dollar range but I can
personally attest to the ability of removing the stale odor of
cigarettes from a smokers room while utilizing less than $10/year
worth of electricity.

Good luck!

Quick answer is that you have to get rid of the odor. As you indicate
in your question, you really can't cover it up with Febreze or
candles.

It's most likely that the odor is caused by mold or mildew which, in
turn, is caused by moisture. You'll have to eliminate the source of
moisture, then eliminate the mold or mildew. Sounds like you've
already tracked down the general area--the closet. It may involve
ripping up the closet; there may be a leaking pipe, dripping toilet
from above, etc., keeping the area behind the wall moist. First, you
have to stop the water.

Then you have to eliminate the spores. If the closet and other
affected areas are drywall, you'll really have to replace the drywall.
(I don't know about plaster; probably not.) Everything should be
rinsed down with a mold/spore killer (diluted bleach will do the job).
Then dry everything out, and keep the area dry. There are little
heaters, not much more than light bulbs, that you can place in closets
to help keep them dry. You might even get some results now from that.
If you've really cured the problem, you won't need any ongoing
devices, but....

You can also rent ozone machines. Ozone is corrosive and harmful,
though it dissipates rapidly. Hotels use them a lot to get rid of
odors (such as smoke out of a non-smoking room). Basically, you turn
them on, leave the room for a couple of hours, then return and the
smell is eliminated. It works well, but wouldn't be effective
long-term in a situation where you have an ongoing moisture problem.

Hope that helps.