I suspect that you have missed the point entirely: silentcoder made the correct distinction between "mass" (an inherent property that depends on the number of atoms etc. in an object and that is independent of where the object is) and its "weight", which in physics terms means the force exerted by that object on something, which is the mass times the local acceleration.
Thus a person with a mass of 80kg standing on the Earth exerts a force due to gravity pulling them down onto the surface, i.e. 80 kg x 9.8 m/s2 = 784 Newtons. But for all sorts of obvious reasons, we just use the shorthand version to say that the person "weighs" 80 kg.
On the Moon, their mass would be the same, because they'd have the same number of atoms in their body. But they'd exert much less force on the surface, because the gravity on the Moon is only 1/6th of that on the Earth. So, they would weigh less. It's at that point that the shorthand way of talking about weight becomes useless.
Take the person and stick them infinitely far from any gravitating body and there would be no acceleration and thus no force, so the person would be weightless, but not massless (same number of atoms still).
Of course, in low Earth orbit, you're right in pointing out that the Earth's gravitational acceleration has not diminished much. However, while you're falling freely towards the surface of the Earth under that acceleration, the spacecraft you're in is falling out from underneath you at the same rate, so you don't exert a net force on it. Thus you're effectively weightless.
(If you're both falling freely towards the Earth, why don't you hit it at some point? Because you're flying sideways at such a high speed that the Earth's surface curves away from underneath you at just the same speed as you're falling towards it, so you never hit.)
But here's another thing. Under general relativity, gravity is much better thought of as a curvature of spacetime and it turns out that the motion of even massless objects (photons) is affected by that curvature (think Einstein, Eddington, etc.). Indeed, given a very strong gravitational field / very high spacetime curvature, e.g. around a black hole, photons can go into orbit. This is because while they don't have any mass, they do have energy.
So, in a more correct general relativistic setting, even your basic assertion that "to be able to orbit, you must have weight/mass" is wrong.