A king mattress is the same weight class as a queen but much wider and clumsier, so it is really a size-and-clearance problem. The right number of hands and a plan for tight turns is what makes it doable.
Pro tip: On tight staircase turns, tip the mattress diagonally and lead with the top corner, the same hook motion used for a sofa.
Watch out: Do not fold a king innerspring or hybrid mattress; only all-foam models tolerate folding. Bending a coil mattress ruins it.