This achievement is for beating all four elemental archfiends a second time whilst in the Giant of Babil. You will have to fight all four, one right after the other, with no breaks in between to heal or save. Each boss will have 25,000 HP each and will be stronger than the first time you faced them.
Scarmiglione is annoying as he likes to cast status effects on the party including but not limited to: Sleep, Silence, Blind, Curse, and Slow. However, he is nothing special. Simply hit him with Firaga enough times and he will go down in no time.
Cagnazzo is pretty much exactly the same as the first time you faced him. Use Blizzaga on him. When water is surging at his feet, use a Zeus' Wrath, Heavenly Wrath, or hit him with Thunder so he won't get off a Tsunami. At one point, he will probably retreat into his shell. If possible, cast Haste on Rydia and then hit him with Blizzaga until he pops back out.
Barbarricia will be permanently in Tornado mode and you don't have Kain. Simply either cast Thundaga or have Rydia summon Dragon and Fusoya cast Holy. The only thing you have to worry about is the possibility of her casting Lightning immediately after using Maelstrom. Simply have Rosa on standby with Curaja.
Rubicante is the tough part. Whenever he opens his cloak, you have a brief opening to hit him with Blizzaga before he uses Inferno. At this point in the game, Inferno one-shotted my entire party except Cecil. There is a trick to beating him. When you have the chance, cast Haste on Rydia or have her use a Hermes Sandals. Then, when he opens his cloak, have her select Blizzaga, and then press TAB until you get to Edge and have him Steal once Blizzaga hits him.
It should hit him for 5,000 damage and then Edge should immediately steal from him if you timed it right. If done right, he will close his cloak without using Inferno. Rinse and repeat. I was level 55 when I did this on Hard mode and these strats worked fine for me.