This is an inaccuracy in the King James Version.
The Hebrew word is חַשְׁחִ֨ין (ḥaš-ḥîn), the masculine plural participle of the verb חֲשַׁח (chashach) meaning “need”. This means that an accurate translation of Daniel 3:16 should be similar to the English Standard Version, which has “O Nebuchadnezzar, we have no need to answer you in this matter.”
The word חֲשַׁח is rare, its forms occurring just three times in the Bible. In the other two cases the KJV uses “need”:
Ezra 6:9 And that which they have need of, both young bullocks, and rams, and lambs
Ezra 7:20 And whatsoever more shall be needful for the house of thy God
In Daniel 3:16, the way to make sense of the KJV is to take “careful” in the archaic sense “full of care, trouble, anxiety, or concern” (OED), so that Shadrach and companions are to be interpreted as saying, “we are not concerned [or troubled, or anxious] to answer you”, which is similar to the Hebrew, but not quite the same. Possibly the KJV translators were puzzled that the men said they did not need to answer, but answered anyway, and chose this interpretation to resolve the difficulty.
The word “careful” is used in this sense in a few other places in the KJV, for example:
2 Kings 4:13 Behold, thou hast been careful for us with all this care
Luke 10:41 Martha, thou art careful and troubled about many things