Haha this is lovely! 
If you put a drunkard on that bench, he’ll be an additional weight that will likely keep the bench from flipping. Increase the density on the sign, and you should be good.
The Distance Constraint is just that, keeping, avoiding or maintaining a distance between two points.
I think I see what you want, you don’t want Distance Constraint for this, instead you can use a limit similar to how an elbow and a knee works to make a door hinge.
Something like this?
Here’s how you can do that.
Step 1 - Setup
I’ll make a door similar to the one you have.
Step 2 - Hinge
Next we need somewhere to Reparent the door, this will be our hinge.
Step 3 - Reparent
Make the hinge a child of the door. You can swap these around, but the pivot of the hinge will be at the child rather than the parent, and in my case the pivot of the door is at the center of the door, whereas the pivot of the hinge is at the center of the sphere, which is where I want my door to revolve around.
Step 4 - Limit
Finally, lock all axes (a value of -1) except for the one the door is meant to revolve around, which can have a value equal to the amount of rotation you want to allow the door to have.