Is it even possible to set up a system in 12x12 room?

DO NOT replace equipment. Everything you have is fine. Some people think they need different stuff, which is a fallacy and waste of money. Plus you end up with the same problems as before.

I'm in an 11' X 12' X 8' room. The listening chair is against the back wall. Monitors on stands are 5' 6" from the chair and 56" apart. There is plenty of room behind the speakers and on the sides. The equipment racks are on the (drum-roll, please) front wall! There is a TV there also. Record storage and computer desk are on the sides. Cozy.

Treat the trihedral corners and reflection points on the sides and back wall first. GiK Acoustics will probably start with more than is necessary, but they will be a great help. Talk to John Dykstra, he did a very thoughtful design for my room.  I would do it incrementally because you don't want too much control.
You're starting from scratch so you have the luxury of doing almost any configuration that will fit in that room. Jim Smith (GetBetterSound.com) suggested going to the 45 degree setup, but I would have to tear my room down and start all over. For me the treatments were a better option but if I were doing it from scratch I'd do more homework to design the room correctly from the floor up.

It took a long time to get where I am now and if I had been more thoughtful when I moved in to the room it would be laid out differently and I would have had way less trouble. I thought I would never get it right. I had two modes that were annoying but now I can't believe how good it sounds.

Now I'm going to install two REL T/7i's. They say they will work although I am skeptical. Proper room treatments can make a kids phono sound great (almost).


Rollin

I’ll keep it short... what works for me in my 8 x 12 room. Two "tricks" that made all the difference:

1. Turn down the bass (a lot.. below 150 hz or so) in one speaker only. this reduces the boom effect caused by close room corners in your small room. Bass is non-directional and will sound so much cleaner. (less corner effect, fewer phase problems in small rooms). Try it!

2. Judicious use of a separate channel equalizer. Necessary to reduce inevitable upper midrange peaks (shrill) exacerbated by close-in walls, even with treatment (especially at higher volume levels) as well as provide the bass control mentioned above.

A smaller room can cause greater differences than amplifier and even speaker selection as you know. Let me know how it works for you!

Or you could go digital and put in a DSP and never look back. It kind of eliminates the analogue part all together. :)

I have a perfect 12x12 room as well that I use as my main listening space.  I had terrible luck with it when I moved in ~10 years ago but now have very good sound.  Some things that worked for me:

1. Listening on the diagonal.  Set-up speakers such that you are facing a corner.  The best things about this way is that you will get incredible center fill that is deep.
2. Bass traps in the corners (I do two corners, floor to ceiling with GIK).  If you have slap echo or such (clap your hands, does it reverberate or sound metallic, or does it sound natural) as I do, I would but some additional treatments in the ceiling-wall junction. (I use Auralex LENRD bass traps with push pins to hold in place.)
3.  Play extensively with speaker and chair position.  You will find that bass will vary dramatically depending on speaker and especially chair position.  With my current speakers, I have them nearly pushed to the wall and I listen in the near field such that my chair is almost dead center in the room.  This worked less well with my previous speakers that needed some distance to integrate.

I had to try a few different speakers as my room is on the bright side even with the treatments.  Any brightness in the speaker choice (or other upstream choices) will be amplified.

Best of luck!

I am all analogue, FYI.