If you underclock it to something like 700 core and 800mem it should run ok in a micro-atx case or mini itx case. Something like a shuttle or micro atx case.
Cooling will be a concern so i doubt it can run at stock speeds in a crammed case.
Have you considered a "oldschool" desktop case? I converted my tower to a desktop.
You need a tower case with 4 dvd drive bays. Flip the drives sideways, tiewrap them in place. Get some black open cell foam (lots of boards are packed in this foam so look in your old boxes) You can also buy filter foam.
Cut it a bit bigger there should be a groove between the plastic front and cage to tuck the sides of foam into so it stays put.
I put my monitor on top of the case that now sits sideways. It saves alot of space. Also the monitor sits at a better height right in front of you.
To do this you need a standard atx tower with 4 dvd rom drive bays.
You can leave the drives sideways, most will work like that, but when you open the drive it can hit your keyboard.
I hate tower cases, a big useless tower case it's clunky and eats up space.
My htpc/gaming system is also set up the same way, with my sony 5.1 reciever sitting on top and a 46" lcdtv.
You might want to consider buying a used hd-4870 since tri-fire is known to rock.