If an object is in free fall and you are moving in the same way (inside or near it), then you too are in free fall and weightless. From an external point of view, you may be accelerating (and even for a circular orbit, you are accelerating perpendicularly to the direction of the orbit), but since the object and the person are being accelerated by the same amount, they will not notice the acceleration.
(To be really accurate, for a large object you can get tidal effects due to the fact that gravity can vary slightly between the highest and lowest point, and you can also get tiny effects due to the mass of the object itself, but these are negligible compared with normal weight).