I'd start with a brand new proper name brand drill bit that is a little under 1/8". And peck at the hole and clear chips frequently. I've had good luck in the past with even small drills over that length drilling true to within a few thou. A very few thou in fact.
But do check that your tail stock ram is centered both side to side and up and down and is not angled to the axis as well. If the lathe isn't true in these respects conical or drifting holes are more than possible.
If that is the case I think I'd drill the center hole in the stock in the mill or even in a drill press. I'd center drill, drill through undersize, ream then flip over and center drill again. Then mount it up between centers using a drive dog and turn the OD details. I'd give the yourself the added length needed to fit a drive dog.
But do check that your tail stock ram is centered both side to side and up and down and is not angled to the axis as well. If the lathe isn't true in these respects conical or drifting holes are more than possible.
If that is the case I think I'd drill the center hole in the stock in the mill or even in a drill press. I'd center drill, drill through undersize, ream then flip over and center drill again. Then mount it up between centers using a drive dog and turn the OD details. I'd give the yourself the added length needed to fit a drive dog.
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