ANSWER: A well for dragons is mentioned only once in the Old Testament. It is referenced by the prophet Nehemiah who was, at one time, held captive by the Persian Empire. He was released and given permission to return to Judah and rebuild Jerusalem and its walls. After arriving in the city, he toured the walls to assess their condition before reconstructing them. On his journey, he mentions something related to dragons.
And I went out by night by the gate of the valley, even before the dragon well, and to the dung port, and viewed the walls of Jerusalem . . . (Nehemiah 2:13, KJV)
A direct reference to dragons in the Bible can be found in at least twenty-two places in the King James' translation of the Old Testament. Some of the more fascinating references include the following.
Their wine is the poison of dragons, and the cruel venom of asps (Deuteronomy 32:33, KJV)
I am a brother to dragons, and a companion to owls (Job 30:29)
Praise the LORD from the earth, ye dragons, and all deeps (Psalm 148:7, see also Isaiah 13:22, 27:1, 43:20, 51:9, etc.)
The vast majority of Old Testament references to dragons are derived from the Hebrew word tanniym (or tannim, Strong's Concordance #H8577). According to Strong's, this word can mean a jackal, serpent, sea serpent, or even a land or sea monster. This same word was used for Moses' rod that became a snake before Egypt's Pharaoh (Exodus 7:8 - 9).
What do we know about this well for dragons found in Nehemiah? The New Interpreter's Commentary thinks it could be another name for the natural spring named En-rogel mentioned in 1Kings1 and verse 9. The International Standard Encyclopedia thinks it was a well dug in this area. An alternative name for this particular location is jackal's well, a translation found in the New American Standard and New International Version translations.
There is another intriguing possibility to consider regarding this well. Henry Morris, in his book "The Biblical Basis for Modern Science," states that references to dragons in the Bible could actually be referring to dinosaurs.
"Unfortunately, because of the reluctance of modern translators to commit the Scriptures to teaching the existence of something they regard as purely mythical, modern versions commonly translate tannim by "jackals" or "serpents" or "sea monsters" . . .
As a matter of fact, if one will simply translate tannim by "dinosaurs," every one of the more than 25 uses of the word becomes perfectly clear . . ." (The Biblical Basis for Modern Science, Chapter 12, section "Dragons and Unicorns")