If we are standing on earth, then some force (gravity of earth) is pulling us down. Can't we make a force which can oppose gravity? Can't we make an object of zero mass?
There is a force that is preventing us form falling through the Earth or through the floor of your house. It is called normal force.
It is not possible in this way to make an object of zero mass, because mass isn't dependent on the force: my mass is exactly the same on Mars or Jupiter.
In our universe, there are only four known forces which are given names such as "Strong", "Weak", "Electromagnetic", and "Gravitation". And, there seems to be an almost continuous report in the news of evidence or rumors of evidence for a 5th force but so far insufficient to be considered seriously.
Of the four known forces, three of them are stronger than gravity. In fact, so much stronger that gravitational forces are ignored entirely during interactions of one or more of the three other forces.
So, yes, there is a normal force that prevents you from falling through the floor. And, this normal force has its origins in the Electromagnetic Force -- that is, the forces generated by charges (static, in motion, and accelerating). Indeed, the Electromagnetic Force is the one responsible for just about everything you can see, feel, touch, lift. Or, let's say everything you encounter, even yourself.
When you fly in an airplane gravity is trying to pull you down to the earth but the Electromagnetic Force displayed in a myriad of ways is keeping you aloft and safely in the air.
You also ask, why we can't make an object of zero mass. Not sure what kind of object you have in mind but I actually do this, in a way, every day of the week. My hobby is amateur "ham" radio and when I transmit I am creating radio waves that have zero rest mass. When my station receiver and antenna system picks up other radio waves of zero rest mass, the electronics of my radio transmute that zero rest mass "thing" into forces that can move and vibrate air molecules (that is, make sound). Yet the only force behind all of this magic is the Electromagnetic Force.