The Bible does not necessarily require us to think of the earth as 6,000 years old. It is true that the shortest period of time that can be computed from the genealogical lists in Genesis would place creation at about 4000BC. However, biblical genealogies sometimes omit generations. The most obvious example of this is in Matthew 1.1 where it says Jesus was the son of David who was the son of Abraham. In ancient days genealogies did not simply trace the chronology of a family. They were used to tell the story of a family. It was legitimate methodology to omit generations not pertinent to the story being told. The author of Genesis was certainly not concerned with giving us the date of creation. He did not have a scientific intent. Therefore, it would have been entirely within his prerogative to shorten the genealogies to suit a different purpose.
On the other hand, as many modern scientists will admit, our dating methods are in no way fool-proof. Many are the examples of tests performed by different labs on the same specimens which produced radically different results. There are instances of dating methods being used on volcanic rock formed within the last 100 years which produced results indicating many thousands of years of age.
One of the curious things about dating methods that purport to discover millions of years of aging is that they depend on a principle that seems to run counter to another dearly held modern idea: evolution. They depend on uniformitarianism. They must assume that forces of nature and physical laws have continued unchanged for millions and billions of years.
The processes of decay the dating methods measure must be entirely uniform and subject to the same forces we find in our world now or the methods would produce entirely unpredictable and unreliable results.
So these dating methods must assume the universe we observe now is operating the same as it was billions of years ago.
Yet there is no currently observable law or process that allows, much less explains, the process of evolution. No law of nature we observe now remotely resembles what would be necessary to take elements in chaos and organize them into life, nor do we observe any law or force of nature that explains simplicity turning itself into complexity (let alone the complexity of life).
In fact, the laws and forces that we do observe in our universe demand the opposite result. Energy always dissipates. Nothing tends toward greater complexity but always deteriorates. If evolution is true, whatever forces and laws of nature that account for its occurrence are not now present.
There has, therefore, not been a uniformity over the alleged billions of years. Scientists cannot have it both ways. We cannot assume uniformity in the process of aging and deterioration for billions of years and insist that processes of evolution were at the same time active.
All of this is to say, there is no reason to believe the earth is 4.5 billion years old. Since the Bible is God's word, and he is the only one among us who was there when creation happened, I prefer to take his word for it. Creation may not have happened in 4000BC, but there's no reason to doubt that Genesis is much closer to the truth than the current state of our science. I do not believe Genesis chapters 1-11 should be accepted in a symbolic sense. I believe they should be accepted in the plain sense of the words in which they were written.