Remathilis
Legend
Not literally impossible of course, but, as you laid out, above, not something the game was trying to do, nor ever did at all well, with extensive house-ruling required to even take a shot at it. Until 4e introduced the Warlord in the PH1 and you could suddenly have a practical all-martial party, followed by inherent bonuses in DMG2, and low-/no-item became practical.
5e's doing it in the opposite order, Bounded Accuracy makes no-item campaigns practical enough, we just need the Warlord, and, really, some more martial options...
...and it would sure be nice to go further with all-martial campaigns than 4e did.
Tangent: I really question how many people are playing no-magic or no-caster D&D; it seems there are so many better suited systems for that, The One Ring being key among them. Seems like all-martial is a small sub-niche, probably right on par with fully-developed Modern rules, but I digress.
That stated, I still think an all-martial campaign could be done using just a few houserules. Were it me, I'd do Fighter (Champion, Battlemaster, Banneret), Rogue (Thief, Assassin, Mastermind, Swashbuckler), Monk (Open Hand), and Barbarian (Berserker or Totem) and then use the nonmagical Ranger (Hunter or Beastmaster). I'd then use the Healing Surge variant (burn HD in combat) to fix the healing problem and only allow ritual casting via feat. (Banning Magic Adept and other Caster feats as well). Lastly, I'd make sure monsters don't have immunity to magic weapons (resistance is fine ocasionally, but I might allow unique bypasses like stakes for vampires) and go with that. (I might do a few more minor edits, like to what races are allowed, but that's flavor). All I really did was add Healing Surges and the Variant Ranger, the rest is all limits. I'm sure between Surges, the Healer feat, and a slight tweak on potions of healing (being nonmagical herbal tonics) I could run nonmagical just with the PHB+SCAG. I wouldn't be running Princes of the Apocalypse like this, but its doable without additional classes. (Though I won't say an additional class wouldn't be useful or wanted, just not needed). Its certainly on par with the amount of changes needed to make 4e itemless...