Albert Barnes Commentary 1 Kings 19:3

"And when he saw that, he arose, and went for his life, and came to Beer-sheba, which belongeth to Judah, and left his servant there." — 1 Kings 19:3 (ASV)

The rapid pace of the original narrative is very striking: “And he saw (or, “feared,” as some read), and he rose, and he went...” Elijah's fear and flight are very remarkable. Jezebel’s threat alone likely did not produce this extraordinary change. Instead, it was probably caused by a combination of factors: partly a physical reaction from the over-excitement of the previous day, and partly an internal turmoil and doubt about the wisdom of the course he had taken.

Beersheba is about 95 miles from Jezreel, on the very edge of the desert et-Tih. Elijah could not possibly have reached it until the end of the second day. The text seems to imply that he traveled both night and day, not resting until he arrived there. It was one of the towns assigned to the tribe of Simeon (Joshua 19:2). The Simeonites, however, had by this time been absorbed into Judah.