"Now when Elisha had fallen sick with the illness of which he was to die, Joash king of Israel went down to him, and wept before him ... So Elisha died, and they buried him. Now bands of Moabites used to invade the land in the spring of the year. And as a man was being buried, lo, a marauding band was seen and the man was cast into the grave of Elisha; and as soon as the man touched the bones of Elisha, he revived, and stood on his feet." (2 Kings 13:14,20-21 RSV)
Elijah and Elisha
Elijah, and Elisha who succeeded him when Elijah was taken away in a "chariot of fire" (in illustration - see Where Did Elijah Go?) were two of the most powerful Prophets of God in Bible History. They did everything from calling down fire from heaven (1 Kings 18:37-38) to raising the dead (2 Kings 4:18-37), but as true prophets of God, their great power was of, and from, God. As even Jesus Christ said of Himself (John 5:19,30), Elijah and Elisha could do nothing of themselves, or for themselves. As the opening paragraph makes plain, when Elisha's natural time to die came, he suffered and died just like any other human - even though during his lifetime the Holy Spirit healed the sick and raised the dead through him. It was the Holy Spirit of God that made it possible.
Elijah was an ordinary human being, who knew hunger (1 Kings 19:6), weariness (1 Kings 19:5), and fear (1 Kings 19:2-3), just like any other. After being threatened by the wicked Jezebel, he "went a day's journey into the wilderness, and came and sat down under a broom tree; and he asked that he might die, saying, "It is enough; now, O Lord, take away my life; for I am no better than my fathers" (1 Kings 19:4 RSV). Neither was Elisha, or any other human before or since.
The Holy Spirit Remained
In the miracle of Elisha's bones (unlike the widespread fraudulent religious industry of "miraculous relics of saints" that has been operating for centuries, right to the present day), for however long it lasted (there is only one recorded miracle from Elisha's bones), a measure of the Holy Spirit actually remained for some time (at least a number of months, since all that was left was his bones) with Elisha's body - the only way that the miracle of reviving that dead man (back to physical life from which he would again have died later - see "Resurrections" in the Fact Finder question below) could have happened.
Fact Finder: Are true believers "sealed" with the Holy Spirit until they receive their inheritance, i.e. eternal life, which will begin at their future resurrection (see Resurrections)? Ephesians 1:13-14