Worms only run properly with a worm wheel, which is different than a spur gear. A spur gear lacks the helical twist to the tooth which makes for a proper mesh with the worm. But you are correct, it is impossible to eliminate all backlash through precision machining and mounting of the gears alone. Worms create sliding friction against the worm wheel which causes some degree of wear immediately, if you attempt to run them "tight" together. Heavy oil lubrication is required, and this, in itself, will create some degree of sponginess to the mesh.
As Al pointed out, there are precision gearboxes with low backlash available, but there is always some.
The best way to eliminate backlash is to place the feedback system somewhere beyond the gear train, such as with a linear scale on the sliding elements, if you are translating rotary motion into precision linear motion. Or, mount a rotary encoder directly on the ballscrew, after any gear train.
Timing belts may have some backlash, depending on the style chosen. Its harder to see it, and is probably negligible on a system that moves freely. But heavy loads can make the belt slip a tiny amount from one side of the belt tooth, to the other.
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