Is it possible to learn to shoot accurately without aiming (from the hip, by intuition, muscle memory or whatever - I mean without using the sight)?
I don’t mean long range of course but let’s say about 10-30 meters or something like that…
Sure.
As you noted, it's not particularly long range.Wonder if trick is in the stance (for example, if he would be able to shot a target that’s aside from him, not perfectly in front of him).
Do a you tube search for Bob Munden. Bob just passed away recently, but he was the fastest hip shooter.
But, if you have plans to try something like this remember that most of the masters of this kind of shooting spent many years perfecting their skills shooting wax bullets. Wax is easier on your leg when the gun goes off before you clear leather. 
I haven’t see it for sale for years, but there used to be a little kit called “Quick (S)kill” that consisted of a couple pairs of goggles, a little booklet and a few aluminum discs ranging from 4" down to coin size. The deal was that they wanted you to put the rifle on your cheek exactly the same way every time and move your whole upper body as a unit. One guy would stand 4-5’ in front of the shooter and slightly to the side and gently toss the 4" disc up in front of the shooter holding the BB gun. Shooter would snap the gun to his cheek, pivot to the disc and hit it. The progression went to smaller discs and farther away until the shooter could hit a coin pretty regularly from 15’ out…something that would have seemed impossible at the outset.
Yes. One of the simplest ways to practice is to close your eyes, draw, and point to where you remember the target being. Open your eyes, look through the sights, and see how far off your sight picture was. Repeat. For man-sized targets at close range it it remarkably effective.
There have been some impressive handgun shooters who have mastered the art of point shooting with handguns. But it is not something I endorse.
Running the can down the road.
When I used to do a lot of shooting we would sometimes take a semi-auto .22, hold it at the left hip (for a right hander) with the left hand and work the trigger as fast as possible with the right index finger. With a little practice we could simulate full auto fire pretty good. With some more practice we could get a pretty good grouping (considering the method). By good grouping I’m guessing maybe 10 shots in a 2ft. circle at 20-30 feet.
The same is true, by the way, for cameras. When you take a real hell of a lot of pics, you can learn to “aim” your camera, even when you’re holding it way out away from your body. Like holding it up high, and aiming down, to get pics over the heads of a crowd, or snapping pics around a corner. You begin to have an instinctive feel for where the camera is aimed.
(For some odd reason, I feel obliged to say, “Don’t try this with a gun!”)
Hell, you can try it at home with a $2 laser pointer. Just point at something (a picture, a chair, your cat), and then click it on.
My cousin taught himself from a book in the 70s I have no idea the name. Key was a daisy pump action BB gun with the sights removed. My cousin could hit a soda can shooting from the hip every time at ten paces or so.
I can’t resist mentioning the late lamented comic strip Rick O’Shay, one of the major characters being a badd-ass (but actually a good-guy) gunslinger named Hipshot Percussion, seen in one of the strips shown in the above-linked site (the dude in the black hat, of course).
Additional on-line examples here.
British officers were taught to shoot their service pistols “instinctively” (basically from the hip) in the latter stages of WWI and throughout WWII, in emphasis over more deliberate aimed, target-style shooting. The requirement was to put a bullet into a 16" x 12" “centre mass” target at about 10 yards and most people were able to accomplish that consistently with moderate training, apparently.
The book was very likely US Army Manual TT 23-71-1 “Principles of Quick Kill”. I trained with this method and became reasonably accurate in point shooting with a BB or small caliber rifle. The trick (aside from a whole hell of a lot of repetitive training) is maintaining the same stance and upper body position, and essentially pointing with the arm. I was never as accomplished with a pistol, though at one point I could keep five rounds all within the 8-ring at 10 meters shooting from a chest high side or low ready front position, which practically speaking is as good as most trained shooters can hit under real life stress conditions.
As Trinopus indicates, an experienced photographer will develop similar skill so as not to have to rely on the viewfinder for action shots, though for any precision or verifocal shooting the viewfinder is crucial to getting the correct focus and framing.
I’ve personally known two people who can hit a hand-sized target with a thrown knife or tomahawk at 10 meters or more blindfolded (after positioning the target visually), which is kind of the ultimate “point shooting” (i.e. proprioception) skill. I don’t know how long they spent honing these skills to this degree or what amount of natural ability went into this, but this certainly indicates that developing the skill to accurately fire a gun without visually indexing is possible for some people.
Stranger
Late 1960’s Marine Boot Camp…we didn’t spend an inordinate amount of time on it, but we used BB guns. The DI would toss up metal discs, maybe 3" in diameter. From a distance of maybe 10’ or so. We held the gun on our hip until disc was tossed. It took several tries, but eventually a good percentage of the platoon got fairly proficient at it. Just a more challenging version of skeet/trap shooting.