What system can offer the excitement, sense of risk, and overwhelming feeling of despair that characterizes berserk?
None. That's all on the players. No system can force a certain type of emotional investment, no matter how hard it tries.
Ravenloft (a D&D setting) probably comes the closest, stylistically, of any published setting.
One other approach is the Cthulhu Mythos parts of Conan. Weird, dangerous, very dark, but beatable by men of courage and skill. (Didn't know that Conan was part of the Mythos? Go read more of Howard's stuff. Conan deals with elder evils and dark gods as much as, or more than, your random fish-man of Dagon. But Conan faces them in a manly fashion, and either kicks their slimy butts or runs away because he doesn't need to fight them, he already did what he came to do.)
So, what I'm looking for is:
A system where combat is common, even necessary.
Combat is deadly.
The enemy outnumbers you, and is in a better strategic position and/or is vastly superior to you in raw power if not ability.
You have to strive to survive the environment, and not a day goes by without encountering some new deadly threat or risk.
Your sanity is at stake as the physical and emotional strain begins to erode your will.
That's not a very good description of Berserk. It may be a good description of how it looked to you, but it's not a good description of the series.
Combat is common and potentially fatal. Enemies outnumber you and tactics are important. Fear and madness are hazards of the road. Check.
The thing is, the enemies stand the usual Pulp chances against the protagonists: get slaughtered or only get to incapacitate the heroes long enough to meet the big bads. They have to come in huge waves to be interesting but those waves are mostly minions, mooks, extras, and scrubs that will be slaughtered by the end of round two. The only challenges are the named evils, and those are beatable.
Due to the nature of predestination, characters literally
cannot die before their appointed time (which remains unknown to them). They also cannot live past their appointed time. Which means they can survive some truly insane things (like sharing a pack of smokes with Cthulhu and then setting Big-C on fire before hitting him with a French fishing trawler; loot that corpse-pile, baby!) as long as they aren't supposed to die yet. Berserk is "DM won't let us die until we reach the big-bad" gaming. Allies fall constantly but the central character(s) move on. Only when you've reached appropriate narrative points can main characters die.
So, my advice is to use whatever game system you want to and focus on the story-telling. You've had some good advice elsewhere in this thread, but ultimately system won't matter much. I could see this campaign being done with anything from vsMonsters to Mutants & Masterminds to Dungeons & Dragons to Savage Worlds to various Call of Cthulhu systems. Any of them could do it, but the combat-oriented ones will do a much better job of it.
Good luck.