Taylor's partial boil answer is a good one, particularly if you are limited to the typical small-sized electric stove found in most apartments. Another issue with boiling wort in an apartment is excessive humidity, and more problematic, the smell drifting into your neighbor's loft.
But if you want to do a full boil, there is an alternative: go outside! Many apartment complexes have outdoor patio areas. With a bayou burner and propane tank, you're portable -- you can brew anywhere, and the heat generated on these portable turkey-fryer-like stoves is hot enough to boil any sized brew kettle. And if you're brewing in a shared space, you're bound to attract attention, and possibly a brew buddy.
Another idea is to brew at your local homebrew shop. Sometimes you will find a brew-on-premises style shop designed for this kind of thing. If that's not available, ask your shop if they will let you brew outside next to the shop -- offer to do a monthly demo brew outside so the shop can attract customers -- a win-win for both of you. They might even offer you a discount or free brewing supplies in exchange for teaching a class on how to brew.
And of course, you can always brew with a friend at his more spacious home. Moving to 10-gallon batches is not much of a stretch, with you carting your half home in a carboy or corny keg, and your brew buddy keeping the other half. Not a bad way to spend a Saturday afternoon, brewing with a friend.