Topping the charts is Gray Fox's heroic moment at the end of the game, where he rescues Snake from REX. It becomes a Dying Moment of Awesome too, as Gray Fox ends up obliterating REX's radome at the end while suffering from mortal injuries.
It gets even better in the remake, as Gray Fox demonstrates just what his cyborg exoskeleton is capable of. Bear in mind, there are spoilers.
Also, even though he's a villain, Psycho Mantis gets one of these in his boss battle when he reads your memory card and comments on the games. During the fight, he reads Snake's mind and cannot be hit unless the player actually unplugs the controller and moves it to another port. Or shoots a statue. In any case, yeah, Mantis has No Fourth Wall.
Just looking at a shortlist of what Snake does+ out-snipe a world class sniper, beat a tank in single combat, and take down a 100 foot tall Humongous Mecha for starters,+ withstand a Mind Rape from the world's most powerful psychic, fight a cyborg to a standstill with his bare hands, outshoot a world-class duelist, outfight a giant with a fighter jet's cannon strapped to his back, and survive a virus designed to target him specifically without even getting a sniffle for seconds you'd be hard pressed to think of which one is his crowning moment in this game.
Hell, Snake beating a tank with grenades is so badass, that in Metal Gear Solid 4 upon Snake's return to the area of the battle, if the player contacts Otacon via the Codec, Snake mentions this battle and Otacon asks how he won. Snake's nonchalant explanation ("I used grenades. Because I didn't have any armor piercing weapons.") immediately gives way to Otacon's utter astonishment, capped with him remarking that Snake is a total badass.
A Real Life example - In 2010, the UK's Official PlayStation Magazine held a poll for the readers to vote for the best PlayStation game across all three home consoles since 1995 - Metal Gear Solid was second, with only Uncharted 2: Among Thieves coming above it.
Real life example: When they were making the Twin Snakes remake for the Game Cube, David Hayter ponied up half a million dollars of his own money to get the entire original voice cast back. That...Is awesome.
In the ending, how Liquid Snake finally dies.
Solid Snake: "...Die."
The REX fight is epic for how immersive it is. Prior to fighting it, you have to watch as Gray Fox dies, while Snake can't bear to shoot his Stinger missiles at the thing. After this is done, you just want to Kick. Some. Ass. And REX gives you a great target.
The final escape in the jeep when Liquid chases you. After Otacon/Meryl yells hop in, you drive down the escape tunnel and what starts is the most adrenaline inducing music ever in the series and with the countdown going on and excitement that you know this the final part of the game, it isn't uncommon for players to shake from how intense the scene is.
Why the hell isn't Liquid on here yet? I'd say the man is Made of Iron, but not even iron survives as much as he does. Let's see, he survives The Hind D blowing up with him in it, crashing in the Hind, REX blowing up with him in it, falling forty feet down off of REX, getting filled with bullets during the Jeep chase, CRASHING the Jeep, and then when FOXDIE finally hits him, it takes him at least five minutes to finally die. Talk about a badass.
Not only that, though, in the very first few minutes of the game, he successfully shoots down a pair of fighter jets during a blizzard - in a helicopter.
Of course, as far as awesome moments for Liquid go, the fact that he was masterminding things from behind the scenes as Miller and getting Snake to unwittingly activate REX for him the whole time.Magnificent Bastard indeed.
A meta example: when you think Rob Paulsen, you tend to think funny characters like Yakko or PJ. You don't think characters like the Cyborg Ninja. His portrayal of him in The Twin Snakes is downright chilling!