This is not an idiom. It is a particularly common instance of the general construction: "as X as Y", where both X and Y have a large range of possibilities. X can be an adjectival, an adverbial, or a quantifier determiner (much, little, many, few) and Y can a clauses or an ellipsis of a clause. The full clause for which "possible" is an ellipsis depends on the preceding verb. "He left as soon as possible" is an ellipsis for "He left as soon as it was possible for him to leave." When Y is a noun ("He left as soon as John."), the ellipsis is still for a clause ("He left as soon as John left."). So if one was going to do something as soon as possible, then one would perform said action as soon as any impeding activity was no longer an obstacle.
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