But first, a reader question: How much space does it take to brew beer? A few commenters answered the question, but I want to reiterate: it takes about as much room as it takes to store 6 cases of beer.
Most of your supplies get tucked away in storage when not in active use. The bulkiest supplies are the two 5-gallon buckets and whatever bottles you use for you beer. If you have multiple batches going at once, then I guess you should double that. You'll have a big pot (probably 3 gallons) for the actual cooking, and the rest is just tubes and such that easily fits in the buckets.
So you'll need room for 2 5-gallon buckets and 6 cases of beer. That's the answer. I will caution though: it's possible to have a beersplosion take place. Rather than rampant alementals, though, a beersplosion takes place when your fermenting beer is producing extra gases. If those cases can't escape through an airlock, your bucket pops open, spewing foamy beer everywhere.
While hilarious, it does cause a mess. I've never actually had it happen, though, since I'm a stickler for airlocks. Any beer kit you buy will have it included.