Is it possible to have an opinion on something and not have a bias in any way? Take the Israeli-Palestinian conflict for example. If I were to research both sides extensively, learning every single fact I could from every single possible source, then find that one side was clearly in the wrong most of the time. Could I say "Israel is in the wrong more than the Palestinians" without having a Palestinian bias? And furthermore, what does the word "bias" entail? What is its meaning?

I haven't checked my dictionary but I take it that a 'bias' is a favouring of one party over another that is not based on the facts of the matter - it is akin to a prejudice against one party. So I think that if you could discover that X had behaved wrongly more times than Y the judgement that 'X is more wrong than Y' would not show a bias towards either side but would simply reflect the objective facts of the matter. Thus it is possible to have an opinion on something and not have a bias, just in case your opinion matches the facts of the matter. On the other hand with cases of conflict, like the one you raise, it can be difficult to be sure that you really do have all the facts to hand and also that you have an appropriate way of 'weighing' wrongs against each other (is a suicide bomb which kills a child worse than a missile which kills ten civilians or vice versa?). So it may be that in these kinds of cases the charge of having a bias can always be made.