Well I can say it is possible, but not with the same level of control as the example you provided…especially the motion.
The motion in your example has a lot of character, this is most obvious when the sphere make contact and is able to hold onto the tip of the rectangle.
Now as far as doing it, I like the dynamic IK approach as you can still get some control over it, what would be the issue is having the sphere dynamic and a collider body? this should work, although will not provide any force on the sphere.
Otherwise you need to look into the softbody settings…you can use vertex maps to change the strength of the softbody effect, and also the mass…you would want to do this so that you can help the base maintain its shape.
One thing that could make setup easier would be to utilize the mesh-deformer so that dynamics on a low rez cube can be transferred to a higher res cube, saving on calculation / vertex weight painting.
I think in the end, if you only have a few, you could probably animate faster by hand than time spent trying to find a suitable dynamic solution.