Apply Clarke's Third Law
The only way to have technology that can not be reproduced is to incorporate a level of technology that exceeds modern technology by enough to be incomprehensible to those who try to reverse engineer it.
Clarke's Third Law: Any sufficiently advanced technology is indistinguishable from magic.
While this is generally true, I would say it is not absolutely true. The reason that Marvel and DC violate disbelief so badly is because they use Clarke's Third Law as an excuse to ignore science all together and just treat it like another soft magic system. What Clarke's 3rd Law fails to emphasize is that even highly advanced technology is still governed by science. The "magic" of advanced tech is that someone solved an up-till-now unsolvable engineering problem, but the "science" of advanced tech should still agree with generally proven facts.
For example, if you were to go back to ancient Greece with a tank, they world recognize the principles behind the tank. The would recognize that you have a vehicle on wheels, and that your are using metal to armor it. Given some hands on time with it, they could probably glean that than your cannon has something to do with fire and air pressure, but when it comes to the electronics, metallurgy, chemistry, and everything else that goes into making one, they would be missing too many intermediate stages of science between what they have and what you have to recreate it no matter how much time you give them to reverse engineer it.
The other way Marvel/DC violates disbelief is by inventing a new phenomenon as a plot device, and then calling it something that is already well understood to be something else. The number of things these comics call a "black hole" or "gamma rays" can be a pretty big turn-off for anyone with a grown-up level of education. By all means, you should feel free to add some handwavy phenomena, but you should be careful how you label them. It is better to make up something new that utterly baffles on lookers and has no explanation, than to try to explain something in a way that is plainly untrue.
Tropes that work well for this
Super Intelligence: Your setting has a misanthropic scientist who is so far above and beyond normal that he has in a single lifetime gotten too far ahead of modern science for people to understand what he is doing, and he has no inclination of explaining what all those middle steps are. This is generally my least favorite approach since it often ignores the HUGE infrastructure that advanced technology actually requires and creates a terrible return on investment compared to mass produced gadgets. Tony Stark & Bruce Wayne are the only 2 examples of this trope being well executed that I can think of because they were intelligent enough to invent things that were hard to reverse engineer, had access to multi-billion dollar military tech R&D facilities, and most importantly, they both ultimately decided they did not want to sell their weapons technology for ethical reasons. That said, needing to have a private arms company in your back pocket that you spontaneously decide not to use significantly limits the number of heroes and villains your setting can have.
If I were to incorporate this trope as the baseline for all supers in my world, I would have the inventor create a bunch of prototypes as he tries to figure out what production model he will go with, then kill the scientist preventing his work from ever reaching the industrialization stage. Then you scatter to prototypes as needed to fit your plot.
Time Travel: Your heroes and villains are from the future, fighting with what to them is just normal technology. If the current era is ground zero for a temporal war between future civilizations, then much larger numbers of heroes and villains may come into play, but it means that your powers are likely to be standardized kits rather than unique powers. If a single organization sends back 10 heroes, chances are they will all have the more-or-less the same general kits and therefore similar power sets.
Here the future people want to control the narrative of history which they can't do by just giving thier tech away willy nilly; so, they control its distribution very tightly.
Alien Technology: This is my personal favorite because technology falling from the sky can land in anyone's backyard. Since any rando can come across it, you can have an environment similar to Central City of ordinary people being elevated to hero/villain status instead vs everything being accumulated in the laps of the powers that be. This also means no one on Earth actually knows how it works to even try to mass produce it. Lastly, something like an alien ship exploding leaving tech all over the place to be found is going to give you a much more diverse set of powers since the debris form the ship's medical lab will be different than from the arms locker which will be different from the machine shop, etc.
Archeotech: May not really fit into your setting since you are going for alternative modern, but it may still be worth mentioning. Any time you have a massive collapse in civilization the technology to do certain things is lost and what old technologies that remain become the the things of legend. Norse Mythology reflects this in their descriptions of old swords. In the Viking world, many ancestral swords were made using old Roman metallurgy techniques like pattern welding and core piling. By the beginning of the high Medieval period, most of these techniques were lost to time; so, ancient swords were often credited with having magic properties because they were so good. Now, take this concept into the modern world and let's say your world was hit by such a massive solar flare or cyber attack that 99.9% of all digital information is wiped out, we'd be pushed back to the 1950s overnight. Since even our computer factories need computers to run these days, recovery would be a long and painful road... but a few things survived this event; so, your best tech in this world would actually be the stuff too old to reproduce rather than too advanced.
Designing Your Clark Tech
Much like the tank example from before, you will want to always include a few elements of what we do understand, and make sure the things you add on top of that don't go against it. So, let's say you want to make a futuristic rifle that can cut a building in half. Well, we already have a good understanding of how lasers work; we just can't muster the power to make one that strong or the materials to not melt even if we had the power.
This is where the "magic" is introduced. You give someone a laser gun, but it is made out of something so thermal insulating that it that makes Starlite look like paper mache. Then you give it a miniature fusion reactor for power. The materials and systems that go into this thing are so complex and mysterious, that our science does not even tell us how to begin reverse engineering them, but everything "logical" about the device makes perfect since in terms of kilowattage, conservation of mass and energy, and total destructive potential. Our concepts of science tell us that such a device might be able to exist, but does not scratch the surface for how to make it a reality, and that balance is the key.