I've done it myself down to 0.5mm but that smallest one was just an exercise...if I valued my time, it was a $100 bit of much lower quality than what's easiy bought for MUCH less.
https://www.youtube.com/watch?v=N1aj_3tlQhU
You don't need the lathe (though he uses one to rotate it for easy sanding, a drill chuck would work just as well), merely a stone, a torch, and some stock of appropriate diameter...you only need the lathe if you're starting with a shapeless chunk, and this isn't the stone age anymore!
https://www.mcmaster.com/metals/hardened-undersized-high-spe...
For normal-sized drills, at best you're going to drill an oversized (and likely tapered) hole. When you get into the smaller sizes, you're going to break bits left and right.
For normal-sized drills an everyday drill sharpening gauge should do the trick. That'll stop being useful long before you get down to 0.5mm.
If you're drilling wood or some other soft material, and you've got bar stock of the right diameter around that isn't hardened (i.e. a nail), you can sometimes get away with clipping the end of the nail off with a pair of diagonal cutters and peck drilling. You have to peck because the shank fills the hole so there's no way to clear the chips.