Regardless of how advanced a technology is, the universe still follows its core rules; you can bend them, you can find shortcuts, but you cannot ignore them. Also, magnets don't make gravity stop working, they simply apply an opposed force. That cannot be the case for that planetary cluster:
1- The mass of the outer gas giants is so much that they should simply fall upon each other, there is absolutely NO FORCE IN THE UNIVERSE that can keep such heavy bodies so close to each other in a stable position.
2- Magnets, you say? But gas giants are made by gas (duh...), they do not have a solid metallic core like Earth. And even if they did, only the core would be affected: the outer gaseous layer would still fall to the center, or at best form a uniform gaseous blob around the whole cluster (remember that it's planetary rotation that keeps the atmosphere in its intended place).
3- A magnet so powerful to hold four supermassive planetary cores in place would blanket the entire system with a magnetic field so powerful that no form of technology would be able to operate, and would also attract all ships to crash on it or repel them so hard they would be literaly "shot" out of the system.
I could go on for quite a while in listing all the hopelessly wrong things in that cluster. Just NO. It was made for a (debatable) aesthetic purpouse, and that's it.
