There's a tradition among some gamers to find a game they like, then change it utterly and irrevocably. "This is the fantasy game I want...let me just modify the rules until my houserule index is longer than the rulebook." Or, "This is an amazing world and setting for fantasy, but I think these rules would be much more interesting if I modded it out for my dystopian future SF idea I have."
This list is not about that; it's about pointing out some games that actually provide a sound structure (without modification) for running some alternative genres that you wouldn't necessarily expect to see in the game's advertising. No rules changes necessary! No elaborate system hacks required. This list eschews game systems that traditionally brag about their multi-genre emulation. Here goes:
White Star is Actually a Really Good Mecha Anime Game
White Star Galaxy Edition isn't just a decent OSR space opera game. I turns out there are plenty of rules for giant mecha, their pilots, talking and transforming robots, plucky sidekicks, and any other number of genre tropes that you will find fit comfortably in your own emulation of Gundam, Armored Trooper Votoms or Robotech. White Star has you covered!
Traveller is a Perfect System for Lovecraftian Horror
Although you could cheat a bit by simply grabbing Cthonian Stars, which is an actual emulator for science fiction horror tainted by the mythos, the truth is that Traveller has all of the necessary bells and whistles to pull this off. Indeed, the fact that Traveller traditionally encourages an extra or customized statistic by race allows you to effortlessly add a Sanity score to your UPP without ay need to modify the mechanic. The rules for creating alien denizens are just another fancy way of creating terrifying flora and fauna in your mythos-tainted Travellerverse.
Call of Cthulhu is An Ideal Play-It-Straight Historical Emulator
Not only does Call of Cthulhu have nearly endless supplements focusing on different locales and time periods, it's designed to be a realistic game from the ground up, and there's a general agreement among most CoC Keepers that the best way to run the game is to "play it straight" for as long as possible before introducing the horror and ramping up the sanity meter. Indeed, if you want to simply eschew any real need for anything other than the horror of the human condition, in any time period you like, Call of Cthulhu has got you covered.
13th Age Is a Secret Super Hero RPG
The core conceit of 13th Age is "larger than life fantasy heroes who are at the center of the story," and it works exceedingly well. You could, with no actual mechanical changes, reskin the base classes and create a perfect emulator for super hero gaming. Seriously! Take a look and think about it. Weapons are less important in objective terms, but rather in effect terms. Abilities in the system are named but not defined beyond their mechanical relevance. The GM's side of the table is ridiculously easy. This might sort of break the rule up top about not changing the mechanics or setting, but if that's a concern, simply play the game straight from level 1 as a game about demigods (and simply remind the players to incorporate something about their demigod status in their One Unique Thing.) It handles this conceit remarkably well.
Zweihander is A Grim Historical Dark Ages Emulator
It is! And it's not even really a hidden trait if you read through the GM's advice in the book. Zweihander appears to be built not only to emulate the grim world of Warhammer fantasy (or a suitable clone), but to emulate the dark ages of Europe in it's most inglorious form. Even the core setting appears to be aimed at a sort of "alternate medieval Europe" which is even closer to the mark than the faux Europe of classic Warhammer. Using this as the core while riffing from a couple Medieval manuals and atlases seems almost trivial. How much (or how little) you want to involve supernatural elements is entirely up to your own taste.
That last one feels like a bit of a cheat, so here's a bonus suggestion:
Dungeons & Dragons 5E can do Post-Apocalypse Just Fine
Go read up on the advanced weapon tech options in the DMG, then look at each character class and modify their spell choices to reflect only spells of a psychic or potentially psychic nature. Only allow magic items and options which could reflect future tech interpreted through the superstitious view of primitive distant survivors, and either pick monsters of appropriately mutated nature, or reskin to your heart's desire....it's actually easier than it sounds. Or just play it straight and run a Swords of Shannara style future apocalypse in which magic is the source (or product) of the end times.
Extra Bonus: White Star is Star Wars In All But Name
It's fairly obvious that White Star will work to serve as a Star Wars RPG emulator in a pinch; I'm considering the possibility of using it for exactly that purpose while I draw on the Star Wars RPGs for source material.