What is the most compact and simple way of making in Minecraft a 3x3 piston door?
The most compact way to make a 3x3 piston door is to make the door out of slime blocks, and have 3 sequential pistons pushing it up and pulling it down. This is achievable in a 3x3 area above, below or to one side of the door (excluding the door frame). There are many tutorials for these kinds of doors. Example.
If you don't want to use slime blocks, and you don't mind having a 3x3 door without the center block, Etho's door design is probably one of the best out there. It opens quick enough that you can sprint through it without it breaking your sprint, and it is quite simple to build. Link.