The designers of Lojban found a way to combine noun, verbs, and adjectives into single-word predicates called brivla that become noun, verbs, or adjectives according to some extremely fluid process that nobody understands except for John Cowana handful of Vulcans and Nick NicholsHouyhnhnms. The creators of Lojban did something I presumed impossible: they created a worse programming language harder to understand than C++!
Because everything in the universe is either an object (noun) or a process (verb), no, it is not possible for a general purpose human language to omit these two categories. Adjectives are the end result of processes (state).
Therefore because everything in the universe is an object (noun), process (verb), or the end result of a process (adjective), it follows that every human language has the lexical categories of nouns, verbs, and adjectives, even if professional linguists may not categorize them exactly like a Latin grammarian would.