One thing for sure is that we cannot use carbon dating to determine the date that the four gospels were written. This is because the original manuscripts (known as the autographs) of the four gospels and, indeed all the New Testament books are not in our possession. The oldest manuscript we have with a fairly well-supported date is the Rylands Papyrus, which is in the Rylands Museum in Manchester, England. I got to see it last year. This fragment of the book of John has been dated to about AD 125. The date does not come from Carbon-14 dating as it is so small, no one would be willing to sacrifice part of the manuscript. The date comes from the type of writing and the material used. It is approximate (plus or minus about 10 years).
Bottom line, we do not know the exact dates the gosples were written. There are several lines of evidence used. One is the earliest dates these books are quoted from. All four gospels are quoted in patristic writings (a technical term which means writings by the early church "fathers.") before AD 100 in books such as the Epistle of Barnabus, the book of Clement of Rome and the Didache. We can say with a great deal of confidence that all four books were in existence by about AD 90 given the distribution of the books in all the churches. Almost all scholars will give a significantly earlier date to the four books, although some put the book of John as late as the 80s AD. A general consensus of conservative scholars puts Mark at about AD 60-65. Some even put Mark in the 50s AD. Matthew and Luke are usually given a date of writing of about AD 60-70 and John AD 70-90. These are obviously rough approximations. Such dates are based on guesses about which authors relied on the others. For instance, it is not unreasonable (though not proven) to think that Mark was a source for Matthew and Luke. Matthew and Luke relate prophecies of the destruction of Jerusalem (which happened in AD 70) which seems to support these books being published before AD 70. John shows evidence of response to gnostic ideas, likely implying a later date of writing. It is also believed that John lived significantly longer than the other gospel writers. The arguments for the date of writing of these books can get rather obtuse. If you want to get a feeling for these arguments, you should pick up a detailed commentary on each of the gospels and consider carefully the arguments of the authors. A good commentary will present more than one theory and the evidence for the different dates of authorship.
I wish I could give exact dates, but to be honest, we simply do not know the dates these books were written. The key question is whether they have apostolic authority and whether they are inspired. The evidence, both from the quality of the books and from the testimony of the early church is that all four gospels easily pass both qualifications.
John Oakes, PhD