You're going to want the following:

At least 3.5k hit points for all your characters
A synergist who can cast barfire
A good medic (i.e. someone with high magic who can cast cure, cura, and curasa at least, and preferably curaja as well, though it is probably unnecessary)
A party who can all turn to the ravager role
A sentinel
At least one commando with launch (yes, the juggernaut can be launched)

You need the following paradigms:

- Sentinel/Medic/Synergist - for protecting/healing your party and buffing yourself
- Sentinel/Medic/Ravager - for when all your buffs are up and you want to heal yourself
- Ravager/Ravager/Ravager - for increasing the Juggernaut's stagger gauge quickly
- Commando/Commando or Ravager/Commando or Ravager - for slowing down the juggernaut's stagger gauge depletion rate, as well as for when the Juggernaut is staggered to maximize damage.
- Ravager/Ravager/Medic - for increasing the Juggernaut's stagger gauge while passively healing your party.

Start the battle in Sentinel/Medic/Synergist mode to fully buff your party, then, once the entire party has been fully buffed, switch to a paradigm with a commando to slow down the juggernaut's stagger gauge deplietion rate, then switch to ravager/ravager/ravager to increase the stagger gauge rapidly. Switch to Ravager/Ravager/Medic when you need to heal a little bit, and Sentinel/Medic/Ravager when you need to heal a lot. Switch back to Sentinel/Medic/Synergist when you need to re-apply your party's buffs.

Once the juggernaut is staggered, switch to your commando paradigm to launch it and deal as much damage as possible to the juggernaut in the time you have.

Once the juggernaut is no longer staggered, repeat the process until it is dead. It has 1.5 million hit points, so it will take a while.

Note that the juggernaut IS vulnerable to some status ailments, but it has the ability to remove status debuffs from itself with Steam Clean - while this is not a bad idea necessarily, it will cause the juggernaut to retaliate with a powerful physical attack.

What do you get for doing all this?

The Juggernaut might drop some components which give decent item xp, and there is a treasure sphere in the Juggernaut's area which contains a Platinum Bangle, which is fairly mediocre. It also gives about 3000 CP - less than the monsters in the area leading up to it.

But hey, at least you killed it, right?