When you return to Hammerlocke after acquiring the ancient weapons from the Slumbering Weald, you'll bump into Raihan by the entrance to the stadium. Once you're all prepared, enter Hammerlocke Stadium. Go over to where Oleana and the League staff are standing. Oleana will explain the situation. Afterwards, enter the nearby elevator.
You'll arrive at the underground energy plant. Without a moment to waste, head west along the corridor. After going down the slope, there will be an arena-like platform ahead. Run across the bridge, past where Hop is standing. There, you'll find Chairman Rose, who--like any bad guy--won't give up without a good ol' fight.
Macro Cosmos's Rose
As the chairman of Macro Cosmos, it shouldn't be too surprising that he prefers Steel-types. A strong Fire-type should be able to melt his team.
First, Rose will send out Escavalier, an evolved Karrablast. This armoured critter is only weak to Fire, but it takes 4x damage from it. It's fairly strong, but incredibly slow. Be careful of Drill Run, which deals decent damage to Fire-types. Not that it'll survive a Fire-type move...
His second Pokémon is Ferrothorn, which is evolved from Ferroseed. It retains the same weaknesses: Fire (4x damage) and Fighting. If using physical attackers, watch out for its Iron Barbs ability, which inflicts damage if you hit it with a contact move.
Perrserker is an evolved Galarian Meowth; it's the first of Rose's pure Steel-types. There isn't much to look out for. It can use Screech to greatly lower your Defence, Shadow Claw to weak Ghost and Psychic-types and Throat Chop to prevent sound-based moves.
Rose also has a Klinklang, which is a fully evolved Klink. It's another pure Steel-type, so send out your usual tin-opener to deal with it. If you give it the chance, it can use Shift Gear to boost its Attack and greatly boost its Speed. But it doesn't really have any strong moves.
Finally, when things are getting desperate, Rose will send out his Copperajah and immediately Gigantamax it. Its G-Max Steelsurge will place Spikes underneath your team, damaging switch ins. It can also use Ground and Psychic Max Moves. At this point, don't be afraid because it's slow and has middling defences.
After bringing Chairman Rose's schemes crashing down, there's still one teensy, tiny problem left: Eternatus, which has gone out of control. Head back towards the corridor, then walk up to Hop to ride the elevator on the left. You'll arrive at the very top of Hammerlocke Stadium--the Tower Summit. Go up the stairs to encounter Eternatus.
This otherwordly Pokémon is an extremely powerful foe. It can use use Cross Poison and Dynamax Cannon for Poison and Dragon-type STAB, Flamethrower to hurt Steel-types and Dragon Pulse when it's feeling nice. During this battle, you cannot capture Eternatus, so simply focus on defeating it. That said, even that's a task in itself.
If you have a fast and powerful Psychic-type, that's probably the safest option. Otherwise, Ground and Ice-types are also great choices. Dragon-types could work, but they have to be fast or they'll get KO'd by Dynamax Cannon or Dragon Pulse. In the event you have no super-effective 'mons, try inflicting paralysis or lowering its Special Attack.
Bring Eternatus's HP to zero and the fight will end, but the war won't be over. As Eternatus will come back far, far stronger...
Max Raid Battle
You'll fight alongside Hop in a 2-player raid battle against Eternatus in its Eternamax form. Do anything you want, because Eternatus will prevent your moves from working. After the third turn, you and Hop will unleash the power of the ancient weapons you found. Afterwards, Zacian and Zamazenta will both join your raid party; that's more like it!
Now you'll be able to attack and damage Eternatus. Honestly, you don't need to do much, because Zacian and Zamazenta can easily handle Eternatus by themselves. Also, you won't control those two, so let 'em do their thing. If you want, you could weaken Eternatus by inflicting it with a status condition or assist with super-effective Psychic, Ice or Ground moves.
Zacian will fight in its Crowned Sword form. It will start with Howl to boost your party's Attack and Behemoth Blade to do massive damage to Eternatus. Similarly, Zamazenta will be in its Crowned Shield form. It will use Light Screen to protect your party from Eternatus's powerful special attacks and retaliate with a beefy Behemoth Bash.
As for Eternatus, if you couldn't tell by its size, it functions like a Dynamaxed or Gigantamaxed Pokémon. For attacks, it can use Max Wyrmwind (Dragon), Max Flare (Fire) and Max Ooze (Poison). Occasionally, it'll cancel your party's stat boosts and abilities before attacking.
Bring Eternatus's HP down to zero and you'll be able to catch it. As far as we can tell, you're guaranteed to catch it, no matter what Poké Ball you use (but this isn't true for every raid you participate in). So maybe pick one that looks cool? For instance, the Luxury Ball suits its colour scheme.
With Eternatus safely in your collection, the world's been saved! Hoorah!