One of the most popular approaches is to use drive cages, like this one: http://www.chenbro.com/corporatesite/products_detail.php?sku=63
My case has 12 5.25" slots, so at maximum I could fit in 20 swappable 3.25" drives at least in the front. I still have 3 more drive slots in the back, so in theory I could fit in 23 drives, without going external.
In my case I currently have only 17 3.25" drives plus 2 BR burners, but with room for two more drives.
It could look like this (not my own system):
Wow, that is AWESOME. I have never seen anything like that in my life!
I used the CM HAF 932 x case and I love it. The 4 supplied fans spin at 700 - 1000 rpm, move lots of air and are very quiet, but I know quiet is in the ear of the beholder.
5 drives through the side and 2 through the front with 4 available slots for burners, fan control panels, etc.