A converted civilian airliner with huge radar on it flying a predictable racetrack pattern is pretty easy to shoot down—if you ever get in range.

The problem is, you’ll never get in range.

You’ll have to go through the gauntlet of fighters looking for targets, who are in all likeliness already know where you are and where you’re heading—because the AWACS told them.
Meanwhile, the AWACS itself is likely flying over another country.
Some people here has been saying “what about this missile or that technology?”
It still doesn’t change the answer. AWACS is very hard to get to even with long range missiles, stealth airplanes, and whatever else.
In wartime, AWACS is used in conjunction with hundreds, if not more than a thousand, other airplanes. Some of those airplanes will be hunting for stealth airplanes or even the airbase they operate from. An airplane that can’t take off is no different than an airplane that has been shot down. Chances are, in a war with NATO, you got far more immediate threats than the AWACS itself.
Also, if a missile can go very far, great. But can the radar pick up something that far though?
A radar is subject to inverse power of 4 law, 2 one way and 2 on return, so a radar that can pick up something more than 400 km away is very big and as about as stealthy as a bullhorn in an office meeting. A ground installation will be prime target for NATO airplanes and special forces the second the operation commences. Put it on an airplane and the airplane won’t be very stealthy.
If you can’t find the AWACS itself, how are you going to shoot the missiles?
Stealth doesn’t make you invisible; it makes you harder to detect, but at some point you’ll be detected. Turning on radar or transmitting using the radio will also make you susceptible to detection. You need to run through an entire NATO air campaign more or less by sight, also avoiding any ground or naval defenses for more than 400 km, and then, if you’re lucky, you might find the AWACS being escorted by fighter jets. And then you do it again in reverse to escape.
Keep in mind that NATO has more AWACS-type airplanes than non-NATO countries have operational stealth jets, so trying to trade 1-for-1 is a losing proposition.
It’s much easier to send in saboteurs to bomb the airplane on the ground or plant a traitor in the midst than doing that.