I would be careful about not getting that fixed immediately, I've come home to a flooded kitchen several times at my last apartment because apparently the previous tenant was fond of dumping large amounts of pasta and rice down the garbage disposal, and it would sometimes back up overflow my sink. I almost had to get my carpet replaced in the adjacent living room area too as a result, but they ended up steam cleaning it thoroughly instead.
I don't think I could live without one personally, because there are some things that I do not like to throw in my trash can because they stink it up pretty bad - such as trimmings from raw chicken when I'm prepping it, or when I'm removing skin from fish, raw onion scraps, etc... I would have to be taking my trash out constantly because the juices from that stuff would tend to leak through the bag into my trash cans, in addition to washing out the bottoms of my cans constantly. Especially since in all the apartments I've lived in, there was never a window in the kitchen so the air didn't circulate much in there and lingering smells were really noticeable.
Once your garbage disposal is working again, you can keep it "fresh" by throwing lemon rind in it and running it for a bit with clean water. Supposedly coffee grinds are actually helpful to flush out your garbage disposal too, so dumping some used we grinds in it and running it for a bit with running water will clean it out. Pouring some vinegar, particularly the cleaning strength variety, and letting it run through it also helps cut down on the odors.