I believe when implemented correctly either of these two methods can practically eliminate backlash and be DIY'd for less than $50 per axis.
I like the idea of direct drive most, shown here as truly direct however step-down reduction could be used to increase resolution or power if needed. It's more about the idea of motion transferred by rolling. The 'infinite gear'. Obviously the load is limited by the friction between the surfaces. Im thinking of a knurled roller on hard flat rubber (cheap and easily replaceable too). Will require some experimenting. The spring drawn represents some preload with the 'idle' wheel below. I believe with the right combination of materials and preload, this could route wood no problem.
The other method is Cable. Im thinking large cable here, about half inch wire rope as the loads we generate would not cause that cable to distort. I prefer the idea of the fixed cable. I didnt show a tensioner on the fixed but it should be used as having the cable correctly tensioned is a must to minimize backlash. Both show a shaft connecting the other side of the axis which should negate the need twin drives.