I was going to post this as a comment to @KevinReid's excellent response, but didn't want to hijack all the comment space.
I work with 180vDC motors regularly, and they (and their controllers) always have some sort of grounding. Grounding the motor case is usually sufficient, as what it is mounted into is typically metal and grounded, but extra won't hurt. For larger motors, it is more commonly seen.
120vDC? 90 and 180vDC are common in industry. 120vAC rectified to DC is about 170vDC, so I assume your controller is dropping this appropriately, otherwise the motor would be over-driven by 170/120 = 42% and live a short life.
The insulation between the brushes and the case, and the windings and case, should be in the mega-ohms range. In fact there is a tester just for this purpose, called a "megger", which measures this isolation. It does this by injecting a high voltage into the wires and measuring how much current flows to ground.
It doesn't happen very often, but if a coil or brush did fail short to ground, the ground path should be able to handle >=150% of rated motor current. If the motor is fused, this will prevent further damage to the motor. But a shorted winding needs re-wound, which is fairly intensive.
You could try a GFCI, but I think this is asking for trouble. As long as the whole lathe is grounded and fused properly, a motor failure shouldn't hurt the operator regardless of whether a GFCI is used.