I used to live in apartment where I had my workshop spread out all over it in fact! For a lathe I had a Sherline 4410, and I can recommend it 110%. It is very quiet and easy to work with. Also Sherline has a huge selection of accessories. Get a long bed even though you don't think you'll need it - won't ever be regretted! Personally I would stick to this size of machine, although I've had a Emco 8 copy in one apartment. Try to stick to softer metals. Steel can be noisy to machine. Sharp tools and good work holding is important!
Blogwitch is correct that many have normal ME size lathes in apartments. Just look around in Europe and you soon realize that is how it has to be. I countries like Belgium, were I often visit, people have very little room to spare for hobbies, and it's the same in the UK I'm sure. I live in Sweden where space is seldom a problem, unless you live in a larger city. Both the Myford lathes and Wabeco's are designed to be quiet. The Wabeco lathes is probably the quieter of them. But I can't afford them so I do not know for sure.
A treadle powered machine would not at all be a dumb idea, in fact it would have the benefit of control and a reverse, over the normal Sherline motor. Could be set up with bike parts, something I consider doing one day actually. It would have to be a fixed installation though, so no more storing the lathe in a closet. I would use a fixie hub and an old steel rim, that I would put a "wood tyre" on.
Stuff that makes more noise than machining are actually filing and hacksawing. Make sure to get a bench that is heavy for this. I made a sitting bench with a pedestal in one end, that I mounted my bench vise on. I straddled this like a horse. It had rubber feet. This solved the problem of not bolting a workbench to the wall (that would transmit noise into the wall). You will want to have a normal work bench as well for assembly work etc. Put rubber matting on that to deaden noise.
Also, you must always be on good terms with the neighbors. Unless you work too late few are hard to be friends with. It is better to know them, and talk to them about your hobbies.