Is it possible to be unbiased? No; everybody has biases about all sorts of things. Is it possible to bracket one's biases for the sake of open inquiry? Yes, it is. In fact, some say that bracketing is essential to study talmud (which is based in text).
More broadly, there are scientific and unscientific ways of enquiring into the meaning of biblical texts. (Being guided by personal inspiration is an example of the latter.) The scientific method calls for developing a hypothesis, devising a test to evaluate it, carrying out that test, and revising one's understanding as a result. (With much iteration, of course.) Bias, such as confirmation bias (confirming what you already "know" or want to be true), is a concern in the field. Yet scientific enquiry has not ground to a halt.
While formal experiments with proper controls over variables are harder to conduct in the field of biblical heremeneutics than in, say, chemistry, this problem is not unique to biblical studies: many fields share this challenge. (When's the last time twins were actually separated at birth and raised under careful control in pursuit of nature versus nurture?) Yet, somehow, research is conducted and vetted in fields such as linguistics, economics, and psychology, even though people have biases in all of these areas too. So clearly it is possible to bracket one's biases. How is bias accounted for? By scientists showing their work and by other scientists reproducing or refuting their results. One has only to review the peer-reviewed journals and conferences in one's favorite fields to learn that this happens routinely.
But wait, you may say -- that's fine for some fields, but biblical studies involves religion, which is deeply personal, strongly-held, and bound up in people's fates for eternity. Surely even if in other fields people can accont for bias, religion must be different, right? Even if we understand that other fields have "religious wars", and even if we dismiss the talmud argument brought earlier, actual religion must be a special case.
But if that were so, if religious bias dominated everything else, then we would not expect people to ever change their minds. Yet, just as scientists do revise their findings and opinions, a recent study found that about 29% of Americans have changed their religions from one to another (that is, excluding transitions to and from "unaffiliated"). 29%! And that's not counting the "none"s! Even if not all of them did so out of deep personal conviction (but, for example, to unify a household), that's an awful lot of people who seem to have been willing to re-examine and alter their own personal religious biases.
If it is possible for people to change their minds about religion (which it clearly is), then on the way to doing so they must have passed through a phase where they considered a hypothesis other than the one they started with -- that such-and-such religion has the right of it despite what they've always believed. If people can postulate such hypotheses for their own consideration, then they can also postulate them in the context of formal study.
Whether people are willing to do so is a separate question.