What does this phrase mean?

It’s an awkward way of saying that I jumped in and answered very quickly.

More idiomatic would be, “Hardly had he asked a question when I answered.”

“Hardly had he put up his umbrella when it began to rain hard.”

The asker had barely finished asking his question when the answerer answered, that is, very little time had elapsed.

I beleive gailcalled has the correct answer.

I think it’s a bit poetic—perhaps Shakespearean. It sounds like a more archaic phrase.

Synonymous with barely and scarcely.

The word order does make it sound recherché (but not much.)

She had barely asked the question when I rudely jumped in and answered it.

HARDLY, SCARCELY, BARELY, NO SOONER (Source)

I would expect to see phrasing like this in novels of at least a hundred years ago and probably longer.

Nearest contemporary equivalent would lack the inversion and might substitute “before”:

  He had hardly asked the question before I answered.

The “no sooner” variant calls for “than”:

  No sooner had he asked the question than I answered.