Dave Sanderson has given you lots of good advice, which I will try not to repeat.
You don't say how *long* the very small diameters are to be. This is important, as workpiece deflection is one of the biggest problems with turning very narrow parts. Varies a bit with material stiffness, but once you get to more than a couple of times diameter the finished part will taper (wider at the outboard end) too much to be any use. As Dave said, you can use a steady, but the conventional type with fingers will almost certainly not close down far enough (you could try making the fingers more "pointy"). Probably a better solution is to use a rose bit (think of an end mill with a hole up the middle) or one of these small diameter turning tools:
This has advantages: unlike a rose bit, which has to be home made and only works for one diameter, it can work for a variety of diameters (you have to make a bronze insert for each) and it is ready made (and it is quite cheap). It also allows the turned part to have indefinite length.
Even a steady at the end won't help if the turned part is more than a few diameters long - the middle will deflect away. The above devices overcome this.
I suspect the use of hand gravers has in part to do with the fact that not all the turning is parallel, with some curved sections etc, but I'm not a watchmaker, ever. Also, AIUI, the tolerances in watch/clockmaking may not be what you think, I believe sloppy (= very low friction) fits are the norm.
The other issue you need to consider is maintaining concentricity. Very often you will need to work on the end few mm of the part, by pushing most of it into the chuck (to minimise deflection) then to work on the next bit you pull it further out, and with most chucks you will be lucky to have it concentric with the bit you turned first. The best solution is to use collets, which are much better at maintaining concentricity - they are pretty standard on watchmaker's lathes for that reason. This is something to look at when selecting a lathe. The best and most convenient system I have used is that on Myford's 7 series lathes, which has 2 MT collets held in place by a screwed nosepiece. Not cheap, and I suspect from what you say that the Myford lathe option will be outside your price range anyway. You can get MT collets with threaded rears which are held in with a drawbar, work for most lathes with an MT headstock; never used these. Collet systems using adapters for ER collets are an alternative, though in my experience the concentricity straight from the box is not as good, you may need to true up the taper on your lathe, not for the faint hearted.
On the question of lathe size, then up to a point go for something larger than your immediate needs. It is, as you say, always possible to do small work on a big lathe, and not possible to do big work on a small lathe. I say "up to a point" as big lathes tend to be much more awkward to do small work on. I have a Myford S7 (3.5" centre height) and a Harrison M300 (6.5" centre height) but would always prefer to do small work on the Myford (the collet system is part, but only part, of the reason for that). In addition to looking at what collet systems work on your potential lathe, it is very helpful if it has, or can be fitted with, a finely graduated leadscrew handwheel, like the S7. Something I really miss on the M300. Thus means you can position the cut to within half a thou, which is difficult to do with the topslide (as (a) the graduations are too small and (b) you have to set the bloody ting to turn exactly parallel in the first place).
For fine work you need to ensure your slideways are smooth but without a trace of slop; this is one area where a cheap lathe will need fettling. A topslide lock (you will have to do this yourself) will aid security. For getting really fine cuts with a topslide (as opposed to one of the above devices) set the topslide to a fine angle so that 1 thou on the dial gives 0.1 thou of cut (but don't forget to allow for the extra movement towards the headstock, DAMHIKT).
One final thought: tool shape for turning narrow parts should be knife types with a sharp corner, not round-nosed. The latter will cause much more deflection force than a knife type; the latter should have a very, very fine radius to reduce stress raising in the sharp corners, but no more. (This is because almost all the cutting force with a knife tool is parallel to the axis, towards the headstock.)
Hope this helps. If you give more detail about the things you want to make, I'm sure the collective expertise here will be able to add more. I myself have just re-activated my interest in RM after a gap of 30 years, investigating recent commercial N gauge stuff, but have not yet got involved in re-engineering any of it.