Nate Silver's book and xkcd have confused this issue but a bayesian does not generally make strong assumptions about a prior distribution.

A Bayesian can make strong assumptions about the hierarchical structure of probability distributions linking the paramaters/priors to the observations. Frequently she will then use uninformative prior distributions (like a really broad normall) on the paramaters of interest to avoid confirmation bias and combat overfitting when doing inference.

The debate between modern frequentists (like Larry Wassermann whose blog I linked) and bayesians (like Andrew Gelman) has more to do with weather analysis done with an assumed hierarchical model is a good idea. Everyone recognizes that bayes' theorem is valid and priors are a usable tool at this point.

If you provided me research that showed, weakly, that using racial epithets did not affect a politicians chances of re-election (in relevant countries and scenarios, etc.), I would not discard my existing prior that it does, and I would be right to do so. But I would keep it in mind for the next round of evidence in the event I could be wrong.

In this case, no professional statistican is going to perform a Bayesian analysis to tell you the probability that making yourself stronger will cause a decrease in your all-cause mortality. No one is going to aggregate that information and tell YOU what YOU need to do, in this case as in so many others. But there is enough information that you should nonetheless, and this study, like it or not, does add to that information, not detract from it.

P.S.: I disagree with using uninformative priors except when they are warranted. This is how alternative medicine masquerades as "Evidence-based Medicine."