I can scarcely disagree with Affable Geek when he says everyone had an agenda. I am also very much in agreement with his assessment of Tacitus, Flavius Josephus and later Christian "historians." Therefore, I do not propose to reiterate the reasons those historians can not be regarded as entirely reliable.
Where I may, with respect, disagree is in the statement that the Gospel writers were all eyewitnesses to Jesus - or even that just two of them were.
It is now widely known that the New Testament gospels were all anonymous, and remained so until attributed to Matthew, Mark, Luke and John later in the second century. Even conservative commentators accept that the authors of Mark and Luke were not present during the mission of Jesus. And as this Wikipedia article states, the majority of modern scholars believe it is unlikely that Matthew's Gospel was written by an eyewitness. Rex Wyler says in The Jesus Sayings, page 119, that Eusebius applied the name John to the fourth gospel, citing Irenaeus, who allegedly heard this from Polycarp in Smyrna, but points out that no evidence from Irenaeus or Polycarp substantiates these claims. Raymond E. Brown says in An Introduction to the New Testament, page 362, that in the last two centuries the majority of scholars have shifted towards the position that John was not authored by an eyewitness. Thus, all four gospels were written by anonymous authors who were not eyewitnesses to the events they wrote about.
Most scholars now believe that Matthew and Luke were substantially derived from Mark's Gospel. In fact, John Dominic Crossan, in The Birth of Christianity, page 110-111, speaks of a massive consensus among scholars in favour of Markan priority, as well as a major scholarly conclusion that the hypothetical 'Q' document was a further important source used by the authors of Matthew and Luke. On page 114, Crossan says that a major presupposition is that John is dependent on the synoptic gospels at least and especially for the passion narratives and the resurrection narratives. The consequence of this is that instead of having four independent accounts of the life of Jesus, we really only have one - Mark's Gospel. Thus the accuracy of the gospel history stands and falls on Mark's Gospel.
Rex Wyler (The Jesus Sayings, page 16) sits at one end of the spectrum of views on the historicity of the gospels, when he says, "A gospel is an uplifting story, not history." This is a more extreme view, that would mean the gospels are less valuable as history than any other available source.
Rhoads, Dewey and Michie say, in Mark as Story, page 2, that scholars argue that a study of Mark, taken by itself apart from any traditions about it, suggests no connection between the anonymous author and the apostle Peter. Raymond E. Brown, whom I have also cited above, says (ibid, page 160) Mark seems to depend on traditions (and perhaps already shaped sources) received in Greek. This, along with strong doubts that Mark was the author of this gospel, means that the tradition that the author wrote down the memoirs of Peter, is unsustainable.
Undoubtedly Jesus was a historical person, and it is not entirely improbable that Mark is based on reliable prior sources, although being in Greek, as Brown suggests, these sources might have been based on even earlier Aramaic sources. This chain of transmission, including the priority of Mark in the other gospels, requires us to conclude that there are inevitable historical errors in the gospel story. Each stage of transmission from an Aramaic original, assuming there to be one, through the Greek sources used by Mark and then to the other synoptic gospels and finally to John provides the opportunity for error or redaction of the earlier source, and we do indeed see differences between earlier and later gospels in passages where one gospel was dependent on the other. Not everything we read in Matthew and Luke comes from Mark or Q. For example, the genealogies in Matthew and Luke are unique to each of those gospels respectively. Brown points out (page 236) that there are many differences from one genealogy to the other, saying there are those who think of Luke as having tradition stemming from Mary (despite 3:23!) and others who try to reconcile the two genealogies. He says, while Luke's list may be less classically monarchical than Matthew's, there is little likelihood that either is strictly historical.
Ninian Smart says, in The Long Search, page 108, though we have the gospels, and other elements of the New Testament, it is naive to look upon them as bringing us an authoritative portrait of Jesus. The "biblical Jesus" and the "historical Jesus" are the same person, but that does not make the biblical presentation of Jesus as reliable as once believed.