This blogpost on Technic Bricks briefly explains a few 'brick paradoxes' or dead ends. This particular model seems impossible to disassemble by hand:
How can I remove the axle without damaging/destroying the bricks?
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I reproduced the problem and got the axle out using a sculpting tool (similar to an awl). It's a sharp point made of hard metal. (Soft metal will actually bend, believe me, I tried three tools.)
I inserted the point in the space left by the axle groove and pried the axle out. The point of the tool was against the axle while the body of the tool was pushing against the beam's rim. I made a short video to explain.
After looking at the pieces, I saw that it did however leave some damages to both pieces. On the axle it is barely noticeable, While the rim of of the beam has some little notches. Because of these small damages, it's not something I would do on a regular basis. On the other hand, both pieces are totally usable after the 'operation'.
I once got myself into a situation much like this.
What I did was unfold a paperclip and insert the end of it into one of the four corners of open space of the hole that one of the axles was in. Then, I pulled the part of the paper clip out in such a way as to make the most friction between the paper clip and the axle, causing the axle to slide out a bit. I kept repeating this process until the axle slid far out enough for me to get a grip on it and pluck it out.
The tricky part was not pushing the axle back in when I was pushing the part of the paper clip in for another pull. I had to cause as little friction as possible when pushing the unfolded paperclip in, and cause as much friction as possible when pulling it back out so that the axle would come out with it.
This technique may be longer and more tedious than the others, but it uses an item that's almost certain to be lying around your house, and in the end it got the pieces apart without damaging them in the slightest.
A pair of needle nose pliers might just do the trick, or some sturdy tweezers - however it might leave some scratches on the axle (thus limiting the no-damaging).
I assume there's another 2 axle in the other side stopping the use of a wooden toothpick or similar pushing it through from the other end? The joiner has an inner lip, but is hollow if I recall correctly?
One option that may minimize damage to the axle would be to find a mini chuck (left) or micro chuck (right) (i.e. a very small chuck, like a pin chuck) with 4 collet prongs that stick out far enough from the nose cap and are small enough to insert between the rim of the hole and the axle:
Then you could tighten it down so the 4 prongs grip the cross shape of the axle tightly enough to pull it out. The match between the 4 prongs of the chuck and the cross shape of the axle will hopefully distribute the gripping pressure in a way that keeps the axle piece from being damaged.
This answer will only work if there's enough of the notch in the size 2 axle available.
Use a piece of strong string (like dental floss), and tie a small slipknot in one end.
Using some other tool, such as a bent paperclip, push the lasso into the brick's hole down to the notch.
Using that tool, keep pressure on the knot area while you pull the slipknot tight. You don't want your pull to pull the string out.
If you successfully lasso the axle, you then be able to give it a straight tug out.
This should cause no damage to any parts.
I have a small pair of bent-nose needle-nose pliers in with my Lego and use them quite often, generally without damaging anything.
That picture is a 150mm set, mine are a "miniature" 80mm set designed for electronics work. One key feature is that the ends of the jaws have little teeth, so they get positive grip on really tiny things. Of course, if you squeeze too hard you will mark plastic Lego pieces, but the bent nose helps as the jaws flex a little. I found bent nose works better than straight nose as I also use them to rescue small parts from inside big models. For that a long-handle set of pliers is also helpful, but those normally live in my electronics tool kits.
For the above puzzle, note that you should grab one side of the + axle, not the diagonal - the latter means all your force is going onto the edges of the plier tips and you will mark/damage the Lego.
(Echoing Zhaph - Ben Duguid above, but I actually do this rather than theorising about it, plus I wanted to include an image)
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