Psalm 19:4-6 describes the sun as a warrior running a distance. How does the sun get back to its tent in the east? What route does it take?
It is important to keep the general picture in mind. Perhaps the graphic by Ralph V. Chamberlin in "The Early Hebrew Conception of the Universe" (from the Wikipedia page on biblical cosmology) can help:
The sun moves along the heavens during the day. As far as I know, the Bible doesn't talk about what happens to the sun during the night.
However, the first part of Ps 19 is, if I remember correctly, probably an ancient sun hymn, that is, from a related culture in which sun, moon and stars were deities. We know from these cultures that the idea is that the sun makes a same half circle under the earth during the night. In some cultures, this brings daylight into the netherworld.
Whether that concept still existed in biblical theology is hard to tell. It may also have been something that people simply didn't think about. But I may be overlooking some relevant passage.