The answer to the question, "Does the Bible say the earth is 6000 years old?" is no. The Bible does not directly give the age of the earth explicitly.
What the Bible does do is give a chronology in Genesis 5 that, according to some calculations (most famously Bishop Usshar who derived that the earth was created on the evening of October 23, 4004 BC ) indicate that Adam lived around 6000 years ago.
This chronology, when coupled with a {literal | plain sense} reading of seven days of Creation, leads "Young Earth" Creationists to conclude the earth is 6000 years old. Any further apparent age is strictly due to God creating the universe as if it had already been in existence longer.
"Old Earth" Creationists (or Theistic Evolutionists) take a less literal view of Genesis 1 - 12, and suggest that the days are either metaphorical or that these verses mean something else. To a YEC, this flies in the face of a good reading of Scripture, but to more secular scientists represents a compromise between faith and perceived observations.
Finally, many liberals read Genesis 1 - 12 (and hence the Creation story) as pure myth and hence not to be read in the same "scientific" terms that modern man would use when discussing the age of the earth.
Thus, the answer to 'How old is the earth?' as a matter of fact is completely dependant on how one reads Genesis 5.