First, you look all over creation for a drill chuck that will hold it ER16 collets won't go that small, and many drill chucks which are rated 0 to 1/4" are really .030 to 1/4". Maybe the smaller series of ER collets will hold tiny drills, I've never bought anything smaller than ER16.
Buy a 10 pack of drills.
Buy a sensitive drilling attachment, which is a spring loaded drive, on which the drill chuck mounts. The drill press then provides the rotary motion, and you apply fingertip pressure to the thrust flange of the sensitive drilling attachment, and can very tenderly drill the hole.
Break through is the most critical moment when using a sensitive drilling attachment. Make sure to back up the part with something to stabilize and slow the exit of the drill point.
CNC is actually quite good for small drills, as the feedrate is very nicely controlled. However, you need to have a good bit of background experience to know what the drill can take for a feedrate, and how deep it can go per peck. Hence the requirement for a 10 pack of drills
What is really amazing is the guys who can make the flutes in those tiny drill bodies.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)