Not all battles came to melee fighting. You had sieges against castles/towns/fortifications. In these, archers are obviously very efficient because close-quarter combat wouldn't happen until the fortifications were broken. You had ambushes or skirmishes where combat was only shooting a few arrows and then retreating. There are plenty of instances where archers are the best or even only option available.

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If it came to close-quarters combat and frontlines engaged, archers would usually stop firing into the melee to avoid friendly fire. But there might be still targets available to shoot at. Infantry rows would be several men deep, sometimes they were so deep that archers could safely shoot at men at the back. There was movement, battalions would move during a fight and this gave the option for archers to target units not engaged in melee with friendlies. And in some cultures, like Japan, archers were also trained with close quarter weapons. Many Samurai used bows and swords or spears. So your archers would just engage in melee fights with their melee weapon.

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Archers carried around 20-100 arrows with them. The standard depends on the exact time and location. But there usually were more arrows available at the supply camp. So if archers ran out of arrows they could just retreat back and get more arrows. Or they would pick up arrows shot by enemy archers. They did not forge hundreds of arrows in advance but thousands of arrows in advance. Historians estimate that the English shot 125,000 to 500,000 arrows at the French at the Battle of Agincourt. So that battle potentially had 1 million arrows in use.

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Accuracy varied a lot from individual to individual. But archers were usually very well trained. In England, someone who couldn't shoot 10 arrows per minute was considered not fit for war. They weren't just peasants with a stick like some pike infantry lines. So their accuracy was rather good. The length of battles varied a lot, some lasted less than an hour, and others lasted the entire day or would even continue over several days.

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Efficient is a difficult word to use in warfare. Archers were expensive but also effective. So a lot depends on the situation at hand. For example, when defending a multi-billion-dollar castle, the cost of individual arrows is irrelevant and you shoot as many as you have. But shooting unarmed peasants probably wasn't efficient. But using archers would also mean fewer casualties on your side. Replacing men tends to be more expensive than arrows, so even against peasants, it could be more efficient to use arrows.