The spinoff shows are explicit in mentioning (if not quite describing) the necessary technology that the original show only implicitly featured: there's something called the inertial dampening field (or inertia damping field, if you prefer) that apparently selectively negates the effects of inertial mass on those encased in the field.
That is, the crew still feels the rotational and translational inertia of their bodies as if they were doing their walkings and turnings on an immobile floor, but they feel none of the effects of the fact that the floor is accelerating at thousands of gees or spinning around a black hole.
How is this done? It would be too much to ask of the starship computers that they should manipulate this SIF so carefully that it would adjust to allow for the swinging of a fist or raising of a finger while blocking the effects of starship movement. More probably, things encased in the field are simply somehow cut off from the rest of the universe, existing in their own frame of motion rather than in the one of the outside universe.
That's basically what already happens whenever mass interacts with mass: one might think that accelerating at 9.8 m/ss downward (after you, say, jump down from an aircraft) would feel like acceleration, but we experience it as acceleration-free movement, or freefall, because the frame of the universe is twisted around us by gravity.
And Trek engineers are masters of gravity. They can create artificial gravity that pulls our heroes towards the decks of the ship. Naturally they should be able to toy with gravity in other ways as well, including doing the trick where mass in extreme acceleration feels like it's in freefall or merely in standard one-gee acceleration towards the deck.
Indeed, we hear (first in the DS9 pilot "Emissary") that gravity or inertia is manipulated by using the most important Trek technology, subspace fields. Such a field can "lower the inertial mass" of the encased object, as Chief O'Brien puts it. This should make it perfectly possible to negate the effects of acceleration or torque, and to allow ships powered by teeny weeny impulse rockets with teeny weeny fuel tanks to reach accelerations of thousands of gees and travel between planets in a matter of minutes. Who knows, perhaps warp drive itself is merely a more energetic application of the very same technology, one where subspace fields are cranked up so much that ships not only become massless to the outside universe (and can thus move at infinite sublight speed, which Einstein teaches is lightspeed), but start to have negative mass (and can thus move at higher than infinite sublight speed, which would be faster than light)?
Really, the technological marvels we see in Trek would never be complete if we weren't shown the ability to negate acceleration and torque. It is, must be, an integral part of the other technologies already demonstrated!
Timo Saloniemi