Thanks for all the replies guys. Keep them coming.
I have a full basement. So I would have to vent out a window well window, or cut a hole in the brick and do like a dryer vet. I wouldn't really want to do that.
My biggest concern was the smell of the oil soaked lathe. I couldn't run a vent 24/7 so I'm wondering if the house would start smelling like oil. My machinist buddy thinks it's a terrible idea,,,lol.
I have a walkout basement, and it does NOT share air with the house unless the basement stair door is left open. Even with that, "vapor management" is an active thing. You have to do it. It doesn't take much. A small fan out the casement window, and crack the window or door at the other end. Or if the wind changes, reverse the fan, vent the other way. It does pretty well. If you're sharing air with the rest of the house (forced air furnace for example), well, whoever is upstairs is gonna know for sure what you're up to downstairs....
You can also pick and choose your chemicals carefully. "Bulk" cans, trigger spray bottles, NO aerosols, that sort of thing. That goes a long way, keeping liquids as liquids on what you're working on, and not atomized into the air.
Get one of those fancy OSHA approved combustibles or flamibles cans with the foot pedal lid, the metal ones for fire safety/spontaneous combustion containment. Prolly never need it for that (Although it's nice to know you're covered if you ever misidentify something that shouldn't be in a rag bucket), but it keeps good track of your oily, solventy rags, towels, wipes, etc. They're not air tight, but so long as you don't move them around all the time, they keep that stuff contained well enough not to have the smells travel. Oil on the lathe won't go far. Sulfurized cutting oils, WD-40, and smoke from the lathe and cutting oils, that WILL travel.
I have come enjoy a cheap throw rug or hallway runner (Whatever's cheap that day) to keep strategically placed in and about the metalworking area. Chips preferentially stick to that over my sneakers (worn out old sneakers, tied to "slip on" tight, heavy duty slippers really).
Give it a shot. As long as you're not hauling big heavy machines down there (yet....), if you can make it work, well, there you go. If it becomes imposing on the rest of your life.... Well, there you go.
Build a bench or buy a tool box? It's gonna be some work either way. Building a bench means building a bench. Most toolboxes are going to be too tall or too short for puttng the lathe on. So you end up removing the wheels and framing around the box... If you're not even sure how this venture is gonna work out... I'd knock out a simple two by four workbench plan from the interwebs, and bolt the lathe to that. It's small enough that it will REALLY appreciate being bolted down to a fairly stiff surface, but that's relative. I short enough that you're not really worried about top tier build designs to minimize deflections, just a sheet of plywood (or even doubled plywood) across the top will do nicely. I really wouldn't try to go all out right out of the gate, It IS a doable thing that you're considering, but like I say, it's an active thing to keep it isolated. How active depends on a lot of factors, but there's only one way to find out.