sinister wrote:
I need to remove some deep-set nails. They're the type without large
heads, and they look like they must have been placed with a nail gun,
because the heads are (say) 1/8" into the wood.

One method I thought of was some kind of drill bit that would drill
out a small circle, except it would have a hole where the nail is so
the bit wouldn't press on the nail itself.

That's called a "plug cutter". Hard - but not impossible - to use without a
drill press.

--

dadiOH
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