19:3 went for his life. Elijah was not a coward (note his one-man confrontation with King Ahab and then with the 450 false prophets), nor was he afraid to die (note I Kings 19:4), but he was discouraged that he still seemed to have no chance of turning Israel back to God, even after his singular victory at Mount Carmel. Jezebel and her religious system seemed as entrenched as ever.
19:3 came to Beer-sheba. Elijah had been at Carmel and Jezreel, far to the north in the kingdom of Israel, then fled 150 miles to the southernmost part of the kingdom of Judah, and Beersheba, and then even deeper into the Negev wilderness, putting as much distance between himself and Jezebel as possible, finally reaching Mount Sinai (or Horeb) itself.