I dont want the construction set on consoles. I just wanted to know if it was possible.
Nothing is impossible, but this is pretty close.
Don't know exactly how the construction set works, but both the 360 and the PS3 use RISC processors so they're not really fit to run professional development tools.
Also, the interface isn't really fit as well, I think it would be ridiculous to build anything without a (decent) keyboard and mouse.
What they could do is create some "New Game+" features where you have stuff like 'custom game mode', or a simplified item and character creation kit, but I don't think they'll bother with that.
What could and should be possible is to be able to freely download and install mods on your console.
I guess that's not going to happen for many different reasons, but not because it's technically too difficult. Pretty sure the official mods are made with the very same construction kit.