via Closer to Nature
I am going to assume that most apartments have some sort of stove. The key here is if your stove will give off enough energy to boil 5-6 gallons of liquid. You can test out a sample batch of water, it may take a little longer than an outdoor propane burner but you are doing this in the luxury of your kitchen so it works out. If you have a stove that can muster it (and most can) you should be good to go.
One of the other issues people seem to have is that they do not think they are going to have enough storage. Well come on nancy, you have a closest then you have enough space to store a few cases of beer. Throw away your 1996 K Swiss classics you have been holding onto for all these years and that box of memories from your cheating ex.
So if you are a beginner then your first step is to read my "How to get into Homebrewing" post and figure out your set up. My point in all of this is that you really can brew anywhere as long as you have a heat source to get that wort boiling.
There may be some smaller issues like fermentation temperature if you do not have a temp controlled apartment. If your place is too hot in the summer to ferment anything other than Belgians then your answer is simple, brew Belgians in the summer and every thing else in the spring, fall and winter. Or the best move that I made at this time was to by a mini fridge for 30 bucks off craigslist and have my own personal fermentation chamber.
Get creative with your storage and brewing procedures and the sky is the limit for how much you can brew. I have made award winning beers on a gas burning stove that also sat in my closest with my Michael Jordan #45 Jersery. Don't make excuses and get brewing.