For example, if I roll a die, it would hardly be rational to believe "it will not come up 5 or 6", despite the balance of evidence being in favor of such a belief.

A Bayesian would not say definitively that it would not come up as 5 or 6. However, if you were to wager on whether or not the dice will come up as either 5 or 6, the only rational position is to bet against it. Given enough throws of the die, you will be right 2/3 of the time.

At the most basic level, the difference between Bayesian reasoning and traditional rationalism is a Bayesian only thinks in terms in likelihoods. It's not a matter of "this position is at a >50% probability, therefore it is correct", it is a matter of "this position is at a >50% probability, so I will hold it to be more likely correct than incorrect until that probability changes".

It's a difficult way of thinking, as it doesn't really allow you to definitively decide anything with perfect certainty. There are very few beliefs in this world for which a 100% probability exists (there must be zero evidence against a belief for this to occur). Math proofs, really, are the only class of beliefs that can hold such certainty. As such the possibility of being wrong pretty much always exists, and must always be considered, though by how much depends on the likelihood of the belief being incorrect.

If you proposed a complicated belief of 20th century physics (say, Bell's theorem) to Archimedes, he would be right to say he has no evidence in its favor.

If no evidence is given for the belief, of course he is right to reject it. It is the only rational position Archimedes can take. Without evidence, Archimedes must assign a 0%, or near 0%, probability to the likelihood that the 20th century position is correct. However, if he is presented with the evidence for which we now believe such things, his probability assignment must change, and given the amount of evidence available it would be irrational to reject it.

Just because you were wrong does not mean you were thinking irrationally. The converse of that is also true: just because you were right does not mean you were thinking rationally.

Also note that it is a fairly well known fact that 20th century physics are broken - i.e. incorrect, or at least not completely correct. We simply have nothing particularly viable to supersede them with yet, so we are stuck until we find the more correct theories of physics. It would be pretty funny to convince Archimedes of their correctness, only to follow it up with all the areas where modern physics break down.