I use a bush steady for this.
It is a form of (home made) travelling steady, but instead of fingers it has a hole into which you clamp a slice of brass bar as a sacrificial guide bush, drilled and reamed in-situ for each job.
The precision of the resulting rod does depend on starting with reasonably round stock.
Turning stock this small is best done at top speed, but then the length you can machine at one go is limited by the risk of whirling. I have found with mild steel for 6 BA (0.110"), the maximum projection from the bush is about 4".