Yes, there are several ways to determine mass of objects while removing or minimizing the effects of gravity.
First, let's correct a mistaken assumption in the question. Gravity is not the same everywhere. The standard value given for gravitational acceleration is an average of
If you use a spring scale, you'll need to adjust the calibration if you move it to a different location. A two-pan balance compares your unknown mass to the mass of a known object and does not need any adjustment. The weight of both sides of the scale increases with the increase in gravity.
An inertial balance can measure the mass of an object in a way that is independent of the local gravitational acceleration.
In the image above, a flexible steel strip is able to move back and forth in the horizontal plane. The frequency of its oscillation will depend on the mass of the object attached to the end. It doesn't give you a simple number that you can read off a scale. You need to figure out how quickly it moves back and forth and calculate the mass with this information.
Astronauts use an inertial balance to keep track of their mass in space. See this video: How do you weigh yourself in space?
It is also possible to measure the charge to mass ratio of charged particles by observing how the move in a magnetic field. If you know the charge of the objects, it is simple to use this measurement to determine the mass. This technique works well for things as large as molecules. It can be used for tiny drops of water or oil. And it may be practical for things as large as a pin head.