What would a person experience in a free-falling elevator in a shaft long enough to reach terminal velocity?
That is exactly right. A fundamental tenet of physics is that all inertial reference frames are equivalent and indistinguishable.1 Furthermore, given one inertial frame (standing at rest2), any other frame moving with respect to it with a constant velocity is also inertial. The frame "moving at terminal velocity" is just as inertial as "sitting still" and so you would not even be able to tell you were moving.
By definition you feel no acceleration at constant velocity. Thus the acceleration due to gravity must be exactly balanced by some other force. By construction that force is not air resistance for you (as would be the case of a sky diver at terminal velocity) but simply the normal force of the elevator floor, which would make the experience feel exactly like standing in a non-moving elevator in the same gravitational field.
1 At least locally, meaning that any experimental apparatus and things you measure are confined to objects also in that frame.
2 To be pedantic, standing "still" in a gravitational field is considered inertial in Newtonian mechanics but not general relativity. I am speaking in Newtonian terms here, but the conclusion would be just the same if analyzed with the machinery of GR.