Many people believe that the Bible teaches that the earth and universe are only several thousand years old. Although the Bible never actually provides such a figure itself or a specific date from which we could calculate it, this conclusion is derived from adding two figures together: 1) the passage of time from the universe's creation to the creation of human beings, plus 2) the passage of time from the creation of human beings to some undisputed historical event (like the Jewish exile to Babylon or the life of Jesus).
The first figure is usually arrived at by adding together the days of creation presented in Genesis 1. Human beings were created on day six, and so only five days had transpired between the creation of the universe and the creation of human beings. Thus, the days of creation should be understood as calendar days or solar days or human days or "normal" days. Unfortunately this is based on a very superficial understanding of the text. There are several factors present in this chapter and throughout the Bible that render it more complex and much more difficult to maintain the calendar day view.
1. Ancient Hebrew, like most ancient languages, had a much smaller vocabulary than contemporary English. Thus, Hebrew words generally had broader semantic ranges than they do in English -- that is, they were used to refer to a more diverse number of concepts than their English counterparts. This is true in particular of the Hebrew word for "day", yom. Lexicons give this term four definitions: i) daylight, ii) the period of daylight (from sunrise to sunset), iii) a calendar day (24-hour period), and iv) an undefined period of time. These all qualify as literal definitions. If we want to limit the meaning of yom to its root definition (ignoring the fact that this is an exegetical fallacy known as the "root fallacy"), then we are stuck with only the first definition: daylight. Not the period of daylight, not a 24-hour period. Moreover, there was no other word in ancient Hebrew that could refer to undefined periods of time. I've seen several alternate words suggested, but upon examination they cannot function as yom does in Genesis 1. This, at least, certainly opens the door for understanding the days of creation as something other than calendar days.
2. The account of the first day of creation (Gen. 1:3-5) involves the introduction of light and its distinction from the darkness that had previously been ubiquitous. "God called the light 'day,' [yom] and the darkness he called 'night.' And there was evening, and there was morning -- the first day." Again, on the surface this seems to be referring to a normal calendar day, but this doesn't stand up to scrutiny. For one thing, the ancient Hebrews measured the day from sunset to sunset; a new day began once the sun was down. Yet here the text says, "and there was evening, and there was morning," ending the first day of creation at sunrise instead of sunset. This is not how the ancient Hebrews measured the calendar day, and this would have been immediately evident to the original audience.
For another thing, the author changed terms right after defining two sets of terms that would have served his purpose much better. He had just used two pairs of terms to refer to the elements of a 24-hour period: light (or) and darkness (choshek), and day (yom) and night (layelah). Then, when he concludes the first day of creation he specifically uses two different words: evening ('ereb) and morning (boqer). If the first day was intended to be understood as a calendar day, why not use one of the two sets of terms he had just used to define the elements of a calendar day? Why not say, "and there was light and there was darkness -- the first day"? Or, "and there was day and there was night -- the first day"? Instead he goes out of his way to use different vocabulary. This suggests that the author was specifically, albeit implicitly, distinguishing the first day of creation from a calendar day.
In fact, why have this clause at all? Why not just write, "God called the light 'day,' and the darkness he called 'night' -- the first day"? Instead he includes another phrase ("And there was evening, and there was morning") that specifically changes the vocabulary and does so in a way to make the first day incompatible with the Hebrew calendar. The point being that if the syntax and grammar of a passage is particularly unusual and out of the ordinary, it probably wasn't intended to be understood in the usual and ordinary way.
3. Genesis 1:14 states that on the fourth day of creation God placed the sun, moon, and stars in the sky: "Let there be lights in the expanse of the sky to separate the day from the night, and let them serve as signs to mark seasons and days and years." There are several things one could say about this -- for example, the sun, moon, and stars would have been included in the merism "heavens and earth" that were created "in the beginning" -- but for now I just want to point out that this passage introduces "seasons and days and years" as calendar concepts. As such, it is enormously difficult to argue that the three creation days preceeding this were calendar days. If the concept of the calendar day is not even introduced until day four, then obviously the first three days of creation are not meant to be understood as calendar days. Moreover, it is at least doubtful whether the ancient Hebrews could have even conceived of a "sunless" 24-hour day. That is, their conception of a 24-hour day was (in all likelihood) inextricably bound up with the role the sun played therein. In other words, a 24-hour day meant a solar day. So, again, since the sun is not introduced until the fourth day of creation, it is difficult to claim that the first three days of creation should be understood as solar days. At the very least, we would have to conclude that the first three days of creation were radically dissimilar from calendar days or solar days.
4. On day six, God creates various kinds of animals and then human beings, both male and female. Since Genesis 1 tells us that God created both man and woman on the sixth day, all of the events described in Genesis 2 between the creation of Adam and the creation of Eve took place during day six. Genesis 2 states that God placed Adam (Hebrew for "man") in the garden to cultivate it. Then God said, "It is not good for the man to be alone," and so brought the various animals to Adam for him to name. Since none of the animals were a suitable companion, God put Adam to sleep and created Eve. Now if we take this description literally, as those who argue for a young earth are wont to do, it is very difficult to see how Adam could have accomplished all of these tasks during the daylight portion of a calendar day. God placed him in the garden so that he could cultivate it; it would be unusual for God to give him this task and then give him another one before even starting the first. So the text implies (but does not demand) that Adam tended the garden. Moreover, God created the garden specifically for Adam -- not to mention the fact that it was paradise -- so he would have been quite content in it. But after God gives Adam his working orders, he then says, "It is not good for the man to be alone," clearly implying that he was not content. So probably, but not certainly, Adam worked in the garden long enough to grow discontent due to his lack of companionship.
Then God brought the animals to Adam to name them. We can certainly limit the number of animals that he had to name to some extent: he probably only had to name the animals in the garden, not the whole world; he may have only had to name more general groups rather than each individual species; etc. However, in the Bible, as well as the ancient world in general, the act of "naming" was not a simple, cursory process, but rather one which was intended to characterize the nature and essence of the object being named; as such, it required considerable thought as well as extensive knowledge of the object. Moreover, the name was meant to reflect the relationship between the object and the one doing the naming, and this presupposes the passage of a sufficient amount of time for the one to have a significant impact on the other. Therefore, the idea that Adam could have named the (at least) hundreds or (more likely) thousands of animals that God brought before him within the daylight period of a 24-hour day is ridiculously implausible.
Then God put Adam to sleep and created Eve. When Adam woke up and saw Eve, he exclaimed "happa’am," which Brown-Driver-Briggs defines as "now, at length." But if Adam had only been created a few hours earlier, he simply hadn't existed for a sufficient length to justify saying, "Now, at length." Therefore, he had been without companionship for longer -- longer than just the daylight portion of a 24-hour period. But since both Adam and Eve were created on the sixth day of creation (according to Genesis 1), the sixth day was not a 24-hour period.
5. The creation account concludes with the seventh day of creation in the first few verses of Genesis 2. Interestingly, however, the seventh day does not conclude with the "evening and morning" motif as do the other six days. In the ancient synagogues the Hebrews would read the account of the seventh day in Genesis 2:1-3 together with Psalm 95, which warns its readers not to rebel against God like the Exodus generation did, and quotes God as saying, "So I declared on oath in my anger, 'They shall never enter my rest.'" The fact that they read these passages together implies that the ancient Hebrews understood that God's Sabbath rest (as described in Genesis 2:1-3) is the same rest that the Exodus generation failed to enter (as described in Psalm 95); and since the readers of Psalm 95 were warned that they could fail to enter this rest as well, it was an event continuing up to the present time.
When we get to the New Testament, we find this connection stated more explicitly in the epistle to the Hebrews (3:12-4:11). Here, the author expounds upon Psalm 95 by reiterating that the Exodus generation failed to enter into God's rest, and that this applied to the Hebrews living at the time Psalm 95 was written as well (King David's time). According to the author of Hebrews, this means that the rest God offered them was not simply entrance into the promised land of Canaan, because when Psalm 95 was written the Jews had been living in the promised land for many generations (4:7-8). The author further maintains that the promise to enter into God's rest (and the potential for failing to do so) applies to us today (4:1), and that this rest began after God finished creating the universe, thus demonstrating that this rest has been going on for all of human history (4:3b), and is still continuing to the present (4:6). And just in case this isn't clear enough, the author of Hebrews refers to this rest as the seventh day, and quotes Genesis 2:2 to emphasize the point (4:4). Therefore the seventh day of creation was not a 24-hour period thousands of years in the past. We're in it. This is the seventh day of creation.
There are many other arguments I could give along these lines and there are also plenty of counter-arguments one must deal with. From my examination of the issue I have concluded that the days of creation were not meant to be understood superficially as calendar days (or human days, solar days, "normal" days, or whatever). Instead, my inclination is to accept the "day-age" view which holds that each creation day refers to an undefined period of time, but there are plenty of other options on the table. Therefore, we cannot ascribe a date, even an approximate date, to God's creation of the universe from the Bible because the Bible gives us no basis for assessing how much time transpired between the creation of the universe and the creation of human beings.
Update (27 July): See also part 2 and part 3.