Accelerometer to track relative position in space

Would it be possible to use an accelerometer to determine an objects position (milimetrically precise) in space, relative to a know point?

The reference point would be just a few meters away from the device

I mean, would it be that precise?
If I move the device, say, 1mm to the left, would it detect?

Not possible using commercial grade accelerometers. They are too noisy and inaccurate. For more details, see this technical note:

An accelerometer can't tell the difference between sitting upright on the surface and upside-down in the air accelerating toward the earth at 2 g's. To get a position you will need to know the orientation of the accelerometer relative to the local gravity vector and the exact magnitude of the gravity vector so you can null it out. Otherwise your calculations are going to be showing your accelerometer accelerating toward the center of the earth at 1 g.