It's called grouting, and is done all the time in industry
Yes, even big lathes bend, heck, an HLVH or a 10EE (or other super heavy duty short bed toolroom lathe) will bend. How level do they have to be? Depends on the class of work you want to/need to do
Not a lathe, but I met a guy once who showed me pictures of a shop that used to scribe master defraction gratings. Now these things have to be accurate to a fraction of a wavelength of light. TOUCHING a machine meant that it had to temperature stabilize for an addition 2-3 days. The shop was underground, one machine - the machine drew something like 100 watts when running - in the room was a 100 watt lightbulb. After the doors were shut, and the room stabilized with the lightbulb on, when the machine turned on, the bulb went off, machine off - bulb on
The guy I met had an interesting job there - he wrote the software that could predict which ball in the ball bearing was in what spot relative to both races, and compensate for the errors - and they were buying the best ball bearings made