Be'er Sheva is a city in Israel, often considered the gateway to the Negev reigion of the country.
A good place to start hitching in all directions is the Negev Brigade junction, which can be reached by walking three kilometers or so northeast on Derech Hevron out of the city center.
From Negev Brigade junction, hitchhike east to the next major junction, where the road splits north towards Tel Aviv or east (road 31) to Arad and the hotel area of the Dead Sea.
Once can hitchhike to these Negev cities directly from Negev Brigade junction. The road passes through Dimona, meets up with southbound traffic from the Dead Sea at Ha'Arava Junction, and ends in Eilat. Next to Eilat are Jordanian and Egyptian border crossings.