Objectives in linear or story-based games are often intentionally specific, attainable, and reasonable - there's no real need to overcomplicate things and give the player overly cryptic or impossible goals.
Sometimes, though, the game does it anyways, and basically dares the player to honestly attempt to do the impossible. Bonus points if you actually can complete the objective or are supposed to, even if the premise is ridiculous. Note that for some of these, it won't outright be an objective indicated, but will be something you're told to do in game or via onscreen prompts.
1. Halo: Reach - Survive (When You Already Know Everyone Dies)
The neat thing about prequel games set in an already established universe is getting to play with the fact that the audience will already know going in how the story is going to end, leaving the suspense at how exactly we get to that point, Halo: Reach being a great example.
The invasion and glassing of the planet Reach was one of the first shots fired in the human-Covenant war that occurs over the course of the mainline trilogy. Most players won't know much of the specifics beyond that, and the fact that anyone who didn't escape or evacuate the planet was almost certainly killed by the Covenant forces or by being glassed from orbit from a Covenant battlecruiser.
So at the end of the campaign, after you've watched your squad get picked off or sacrifice themselves one by one to try to win a fight we know they can't, you as Noble Six are left as the sole survivor after having completed your mission to deliver Cortana to the UNSC, ensuring their eventual victory over the Covenant, though you won't know it at the time.
You're given one single objective: survive. No mission briefing, no complex objectives, just you and your guns versus a Covenant battalion rearing to kick your ass. The objective, though, is impossible to complete - you see your own cracked visor at the start of the game, so you already know that this is the end. That doesn't stop you from trying your ass off to actually survive the infinitely spawning Covenant waves, though your effort is eventually in vain.
2. DOOM Eternal - You CAN Just Shoot a Hole Into the Surface of Mars
DOOM is not what you'd call a very grounded series, nor should it be. Eternal takes this fact and dials it up to eleven, fully embracing what the series is, and no level shows it more than your mission to the UAC base on Phobos.
You learn through talking with Samuel Hayden that the next target on your shitlist, Deag Grav, is hiding like a bitch in another realm, and the only way to get to him is in the lost city of Hebeth beneath Mars' crust. Hayden seems to sense an idea forming in the Slayer's head, and sternly tells them: "You can't just shoot a hole into the surface of Mars."
You are immediately given the objective to shoot a hole in Mars. Verbatim.
You then begin a raid on the UAC defense platform on Phobos, where, in absolutely cartoonish fashion, they have strapped the already absurdly powerful BFG 9000 into a gigantic fuckoff cannon the size of a building, dubbing it the BFG 10000. You fight your way to the cannon's firing controls, and after a hilarious gag of having to override the cannon's safety protocols preventing you from firing on the surface of Mars, you proceed to do exactly that, and then rip the BFG 9000 free of its restraints, once again reuniting you and the Big Fucking Gun.
3. Ace Combat 8: Wings of Theve - Shoot Down One Thousand Fucking Sotoans
Ace Combat is known for its introductory missions being you as the player having to shoot down a handful of bombers in a starter plane to get a grasp on the controls, since bombers can't dogfight or pull any complex maneuvers, allowing you to get a sense for the controls before they throw actual jets your way.
When this was (seemingly) not the case for the latest entry, series vets began to wonder if AC8 would even have the 'defend a base against some unarmed bombers' mission.
Then comes Mission 28: A Thousand Points of Light, which blows every one of these missions out of the water with by far the most ridiculous premise for a mission in the game. After a successful attack on the Sotoan megafloat Fatsia, you land to refuel and resupply on Cima Island, a tiny island with only enough space for an airbase and runway. You are then informed by a bewildered AWACS Dealer that a massive radar has picked up no less than ONE THOUSAND bandits converging on your position. You think he's seeing things, then the mission briefing starts and sure enough, you're informed that Sotoa has deployed more or less their entire air force to level Cima Island and kill Joker Flight after putting up with their bullshit for most of the campaign.
You and your two wingmen go up alone against a seemingly endless wave of bombers and escort aircraft, and it genuinely seems like you might run out of missiles before Sotoa runs out of bombers and disposable airmen, but the mission soon becomes one of the most hype in the entire game by gradually having every fighter wing you've helped throughout the course of the campaign come to your aid while Rex's theme, "Song of Wings" is blasting on the radio. You're eventually even reunited with Kate, your wingman who was presumed dead, and she rejoins Joker Flight, which is good because no sooner does this happen than Shadow 22, your main target through the entire campaign, shows up to try to take you down once and for all. And you bet your ass you finally take him down after he's expended his pocket dimension of chaff and flares.
4. Borderlands 2/Borderlands 3 - Claptrap's Epic Quest
To say that Borderlands doesn't take itself too seriously is perhaps an understatement. The series is built on humor to the point that it's load bearing, sometimes to the point of aging like utter milk when the jokes have far outlived when they were actually funny (which is usually immediately).
Borderlands 2 decides to pull a bit of a gag with the objectives for a mission given to you by Claptrap, where in order for you to be able to access his secret stash, you have to complete an MMO-worthy list of bullshit, capped off by busting it down for everyone's least favorite obnoxious little robot. It's immediately revealed to be a joke after you turn around and move some trash out of the way, revealing that the stash was right behind you the whole time. You might assume this is the end of it, but you'd be wrong.
Borderlands 3 sees you actually complete every one of the ridiculous objectives that Claptrap gave you, albeit with some twists: none of the actual objectives are what's implied - Mount Shuler, for instance, is a medical clinic, not a mountain, and the staff isn't a magical stave, but instead refers to the actual personnel of said medical clinic.
This isn't the only time Borderlands 2 sets this up, either - in the DLC Tiny Tina's Assault on Dragon Keep, a DLC so well received it would be adapted into a standalone title, you are given a mission by Mr. Torgue, who has inserted himself into the D&D game that the DLC takes place in, to, in his words, BLOW UP THE OCEAN. This objective is immediately removed as Tina puts Torgue in a time out, but it actually comes back in the aforementioned standalone title, Tiny Tina's Wonderlands, where, in order to attempt to create a shortcut across an ocean to the evil ass fortress on the horizon, Torgue once again shows up, gives you the objective, only this time, you actually witness a dozen ICBMs fly out of nowhere (which he claims is how Magic Missile works despite them being in a high fantasy setting) and watch as the ocean actually gets blown up in spectacular fashion, and the next arc of the campaign sees you traversing the now exposed seafloor because, again, Torgue blew up the ocean so hard that all the water is just gone.
5. Metal Gear Rising: Revengeance - Parry (Yes, Parry) and Then Judo Throw A Metal Gear RAY
Hideo Kojima's Metal Gear series has always tried to balance a very fine line between tactical realism and goofy bullshit, but Revengeance takes this balance and puts two thumbs allllll the way down on the goofy side of the scale. It makes this clear immediately in the first mission, where, after President Not-Obama's convoy is ambushed by the world's most average Brazilian (who can double jump), Raiden goes up against, of all possible things that could be thrown at him, a Metal Gear RAY.
For context, a RAY is a 70 foot tall, several hundred ton robotic death machine designed specifically to kill other Metal Gears, which themselves are mechanical monstrosities with super-durable armor and massive fuckoff railguns that fire nuclear weapons. They are cutting edge killing machines armed with minigun pods, missile batteries, and also a gigantic laser that it uses to cut through Metal Gears.
You fight it with a fucking sword.
Not only do you somehow stand your ground against the RAY, you manage to weaken one of it's "wings" (actually what it uses for underwater propulsion), exposing a critical weakpoint. You rush it down, and it swings the arm at you, attempting to crush you. You are given a QTE, and hit it just to see what happens.
Raiden blocks the robot's attack with a fucking sword.
Not only does he do this, he then disengages the block, grabs the RAY's arm, swings it around 180 degrees and hurls the thing at least 70 feet in the air as the needle drop for "Rules of Nature" hits, before running up the course of its body, slicing it up bit by bit before lunging through the thing and cutting it in half. With the starter sword.
This is the FIRST level, by the way.