Paraphrasing Douglas Adams, "Flying is learning how to throw yourself at the ground and miss". This is pretty much how orbits work - you're actually falling all the time, but missing the body you're orbitting.
The reason you're experiencing apparent weight-lessness is because every piece of your body is accelerated the same amount (if you could achieve the same effect when accelerating in a car, you'd have the standard sci-fi "inertial dampeners") - and that means that you're not getting any information about the acceleration acting on your body. After all, you're not feeling the gravity when sitting on a chair - you're feeling the pressure the chair puts on you as it prevents you from accelerating through it. If there's nothing to oppose the acceleration, there's no way you could feel it. Accelerometers onboard would report the same thing - they can actually only measure forces that don't act uniformly on the whole body of the device. If you've got an accelerometer in your phone, you can see this quite easily - as long as you hold it in your hand, it will report a 1g of acceleration downwards. However, drop it (on a pillow or something), and during its fall, it will report 0g (ignoring air resistance - if you add air resistance to the equation, the acceleration will steadily grow from 0g to 1g as the object approaches its terminal velocity).
To be precise, the acceleration isn't actually exactly uniform on your body. The parts that are closer to Earth will be attracted slightly stronger than the parts further out. This is called the tidal force, and it's the reason we've got tides on the Earth (and going further into the past, the reason the Moon always faces us with the same side, and in the future, why the Earth will also always face the Moon with the same side). However, the sensors in your body are nowhere near precise enough to register such a tiny difference (and it really is very tiny on the two meters of the human body a hundred thousand kilometers from the Earth). However, that's only because the difference in the gravitational pull is very small over your length. The fun part starts when you get close to a black hole :)