Your answer lies in definitions.
A historian's job, like that of any academic, is to ask questions, find ways of answering them and present the answers as an approximation of the truth. Historians, like biologists and mathematicians, need to apply objective methodology in their search for the truth.
Gone are the days when you could just write whatever the hell you like using historical figures as characters in a story and call it history. And history is the better for it. One can make a claim but it needs to be evidence based in order to be taken seriously. This is an exciting time to be a historian as there is so much more evidence available to us now. Heck, we're even revising revisionist history these days. Look at Richard III. For decades, centuries even, the narrative of him as the tyrant, hunchbacked king has been challenged on the grounds of a lack of physical evidence to support it and there is plenty of evidence to support a counter-narrative of him as a retrospectively maligned figure, his portraits being doctored post hoc in the interests of strengthening the Tudor/Lancastrian claim to the throne. The tyrant element lacks evidence as well. He's on the way to being completely rehabilitated and then we find what appears to intents and purposes to be Richard III's body and we realise that actually both narratives have some truth to them.
So History needs to be objective in terms of its underlying principles. That much nobody can deny.
History, however, deals with a largely intangible subject matter: human motivation. One cannot perform an empirical study on the past because it does not exist. Even living observers of the past will not produce identical testimonials and physical relics are not representative sources and don't present a complete picture. The answers we supply to the questions we ask are always going to be subjective to our personal views and experiences.
This is not necessarily bad for the subject and a case can easily be made that human motivation is too complex a thing to be explained by a single narrative or a single reason. Take the question of why Elizabeth I never married. When I studied this at A level, we focused on things such as the diplomatic quandary of a female head of state taking a male consort, whether or not she never truly got over her first love, whether she was infertile, whether there was a financial motive etc. Now though, having had children of my own, I would suggest that a large part of her decision not to marry was probably influenced by a fear of childbirth. Mortality rates of women in childbirth were as high as 50% in some populations in Tudor England. If the queen were to die in childbirth, the country would be thrown into chaos and another civil war would begin. If the Tudor line were to continue through a female route - and the only lines left at this point were via this route - better be it some other woman who had already had children and she enjoy her own life as long as possible. Perhaps this explains her reluctance to do away with Mary, Queen of Scots. After all, it's only when MQS represents a direct threat to Elizabeth's own person that she does away with her. That's a version of the truth that's impossible to verify, and certainly represents an incomplete vision of what happened, but it's something that generates further questions of how women were perceived at the time etc.
I digress slightly; the point is, there is no single truth to be found in history because such a thing doesn't exist and never did. Human movement and motivation is infinitely complex. The questions we ask in uncovering it are being continually shaped by our present experience, too. However, the same can be said even of the purest sciences. Likewise, even the purest sciences are subject to our need to present the facts as a narrative, the effect of which will be to distort and skew the truth.
So whilst history - as a reflexive subject - can never be truly objective, this does not have a negatory effect on its value. The subject is simply another degree of dilution of the notion of absolute truth, if such a thing even exists.
I always think a better version of this question would be to posit the view: A time machine would eliminate most, if not all, of the limitations historians currently endure. Discuss. But that's for another day.