You're trying to force a round peg into a square hole
Could you play in a post-modern setting with D&D by just using non-caster classes, non-magical feats, and redefining some of the other classes' features (like the monks ki) to be mundane powers? Probably. Would it be balanced? It's tricky.
Even fighting classes like the Barbarian subclasses come with spell-like abilities (for example, the Eagle Totem Warrior Barbarian has a flying speed from level 14, not something that you can really explain in a mundane setting). So you very likely you would be limited to Fighter and Rogue subclasses that have no spellcasting, and little else. Or you would have to come up with cyber explanations for magical effects.
A game system like 5e is a whole. Yes, there are some core mechanics, like ability scores, bonuses, how skill rolls are resolved and so on. But D&D is not d20 or gurps, or Basic Roleplaying, systems designed, or at least claiming to be mallable, suitable for different genres of games, or malleable enough to support that sort of customization. It is an integrated whole for a fantasy setting with magic. Many features, rules and most exisitingexisting classes do not make much sense any more if you remove magic. The DMG does offer some options for grittier damage, modern weapons and so on (p. 266ff), but that is a long way to a balanced game system without magic. You are going to end up reconfiguring an rebalancerebalancing the entire game, nearly as much work as making a new game.
At that point, why are you trying to do this? Why not use a game explicitly designed for cyberpunk experience, like, erm ... Cyberpunk? I think that this is going to be a better overall experience than trying to make D&D 5e be something it is not, and even with learning a new system, probably less work.