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

We see plenty of systems with the loudspeakers too far from the front wall. Yes, too far. For whatever reason there is a commonly held belief that 2, 3 feet or more is needed for good depth of soundstage. In practice, many loudspeakers can be ideally placed much closer to the front wall with better results in every aspect. Bass, imaging, width of soundstage, and better sound from more listening spots. 

If you would like help just let me know. I can walk you through the process over the phone.

Yes. Also try setting up your system at 45 degrees to a corner 
this eliminates a lot of the dead’s and highs with a square room and setting the sound coming from one end 

Most people never have the 'ideal' imaging triangle, their listening position, like mine and most of my audiophile friends, is further back than the width of the speakers, by a few feet or more. Usually that is combined with the rear wall being farther back than you will have. 

I have equal triangle in my office, and using small B&W bookshelf speakers with a powered sub, it is quite enjoyable. I get just a bit from the sub, you should only notice it when you turn it off.

Your room will force the near perfect triangle. It's problem is square and rear wall right behind you. You will definitely need to strongly reduce reflections off the rear wall. Less bass from the speakers helps.

Using the smaller speakers, toed in a bit away from side walls, perhaps tilted back a bit (not parallel to floor and ceiling), and only going for 40hz, no port is my suggestion. Then, tame the rear wall. Then, if you want, try a small powered sub, but don't go overboard.

Tweeters at seated ear height, bass below that.

Small size speakers and small size sub to enable alternate positioning. NO PORT!!!!!!

I have a small system in my computer room which is about the same size as your room.  My computer desk is right up against the front wall, so I'm very near field.  I have the speakers in the front corners of the room, about 1 foot from the wall and angled towards me at a little less than 45 degrees.  Wood paneling, no treatment (yet, hoping to pick some up this weekend), sounds great!  It doesn't have the sense of scale of my main system, but in terms of sound quality and clarity, it sounds as good, if not better.  Speakers are Harbeth P3ESR and I have a small SVS SB12-NSD sub on the floor next to my desk.

I don’t agree that in a small room speakers always have to be toed in. IME, that is not always the case, and in fact, in my room, I prefer the speakers to be placed straight ahead. (https://systems.audiogon.com/systems/4100) Experimentation is the name of the game here also. The design of the speaker has a lot to do with this aspect.
Also, I do believe that if the equilateral triangle placement can be accomplished, that is preferable.
Lastly, I think that getting the speaker more into the room and off the back wall is going to give you the most depth and imaging precision. Near field listening is the rule in this size of room, there really is no getting around that...which I actually believe is beneficial anyhow.
Rule of thumb...small rooms and small speakers...horses for courses.