I am not writing a story, this just came to me as an idea. How can one introduce superpowers in his world without an appeal to magic or heavy machinery (in fact, without an appeal to machinery which is visible at all)? In some sense, you won't be able to say "look, he has superpowers" unless the targeted person manifests it.
Well, it depends on how you define 'magic' and 'heavy machinery'. And 'superpowers', for that matter. Superpowers, plainly put, are powers that are impossible for a normal human to have. Strength beyond that of a human, flight, speed, etc. In other words, a power that a human can't have without outside aid. It's impossible to have them without the addition of some kind of gimmick to give it to them, which is usually magic or heavy machinery. Take the X-Men, for instance. Some have identifiable mutations (Angel, Beast, etc.) but others are perfectly normal until they use their powers (Iceman, Jean Grey, etc.). Does this qualify as 'magic'? Well, Marvel says it doesn't, because magic in Marvel is clearly defined (Dr. Strange, for instance) and X-Men powers comes from radiation mutated genes, retconned into junk DNA introns, retconned into mutant DNA, but not from radiation. But there's nothing in DNA that would explain the powers, is there?
So if you want to label something as 'not-magic' and explain that's not what's happening, you can do that. But if you want to say that the main characters have unexplainable powers that no one else has without a fantastic explanation? Well, then it's just not a superpower. It's a normal power. Guns, for instance, would be considered a superpower (in the heavy machinery sense) in the 11th century, but a normal one in the 21st. So if you want a legitimate superpower, an ability which is beyond the reach of normal humans, but you want an explanation for how said human has it without resorting to phlebotinum, handwavium, or unexistium? Not possible.
We have historical examples of these occurrence; just to quote some of them:
- when the first European explorers visited the Americas on their horses, the local people took them for divine creatures, hybrid of men and horses, which were unknown to them.
- when the first guns were used against the Polynesian people or other less advanced civilization, a stick which emitted a puff of white smoke and the sound of thunder leaving you dead at a distance was surely perceived as a superpower.
- the cargo cults developed in the Pacific, when the indigenous saw the cargo delivering goods during WWII, just a consequence of a man waving some sticks in the air.
To wrap up, remember the well known statement:
Any sufficiently advanced technology is indistinguishable from magic.
Wolverine style regeneration can probably be implemented with nanites which at our current technology level might as well be magic.
Small scale drones with wifi cameras and microphones could be used to implement super sight, claravoyance and distance-hearing.
Super strength could probably be accomplished through xenografting. There are a number of animals whose muscles have a much higher strength-to-weight ratio than humans. Some of them are pretty close to us genetically. Current technology, animal-rights-groups and rejection issues might present a problem but it's closer-to-real than the nanites.
Extraordinary skill at reading body language when mixed with intuition and good social media hacking skills might pass for mind reading.
Incredible wealth is functionally equivalent to mind control. There are very few who will disobey you if you compensate them super-generously.