Well, yes, that would be the standard way to look at it. Sorry for being confusing.

>Anyway, when you accelerate a car, you are not feeling acceleration, you are feeling the force of the car on you.

According to [Force = Mass * Acceleration] you can take them as equal (apart from the constant factor of your mass). In the freefalling box that is accelerating (and thus no longer in free fall) you'd also feel a force. No need for experiments if you have your normal senses. You'll be pressed against one side of the box. (And if the acceleration is 9.81 m/s^2 you couldn't distinguish it from a box on earth.)

That is actually the basic premise of general relativity - the equality of 'inertial mass' that resists against acceleration and 'gravitational mass' that happens because of, well, gravity. It's like the idea of constant speed of light for special relativity.

edit: Unless you're talking about gravity, which accelerates all parts of the setup equally, thus no force results.. but that would need something else in the experiment, either a great amount of matter or energy. But then you'd have a reference frame right there. (Mathematically. As for an experiment.. hm. Check for perturbances from space-time distortions that result from rotating gravity sources, maybe.. most gravity sources in this universe are rotating. There's an experiment done that checked that for Earth.[1])

[1] https://en.wikipedia.org/wiki/Gravity_Probe_B