One of my physics teachers told me that in the formula of calculating weight i.e. $W = mg$, $m$ is not equal to 0 until and unless you're travelling close to the speed of light. She added, that the acceleration due to gravity, $g$ is also never equal to 0 because for that either mass of Earth has to be 0 or the square of the radius infinite. My question is, if $m$ and $g$ are both non-zero, then how is weight equal to 0?
See the problem with your argument is that Weight is not necessarily equal to mg. If you stand on a weighing machine then weight is equal to the Normal reaction with the machine.
If you stand on ground with no acceleration,then by Newton's Second Law$$N = mg$$ But if you stand in an accelerating lift,you are accelerating and hence: $$N - mg = ma$$ $$=>N= m(g+a)$$
Normal reaction will change depending on the magnitude and direction of acceleration.
In a nutshell ,Weight is the Normal reaction with the ground.
When you are in a free fall , there is no surface to provide Normal and hence Weight is zero and you will experience something called weightlessness.
Ever been on a roller coster?
Here you can actually feel your weight changing similar to that in a lift.
Weight is the force of gravity acting on a mass. If there is a mass in a gravitational field, it will have weight. What becomes zero is apparent weight, it is the reaction force by which you feel your weight. Without it, for example, under free fall you can't feel your own weight and this my friend is called "weightlessness". Reaction force is the force that a weighing machine measures.So if it's not applying any force(reaction) on you, then it will simply show you zero reading. But your weight is not zero, because your mass is not zero.