How can they establish the impact of sports on mortality without control groups? So, this paper in BJM "Associations of specific types of sports and exercise with all-cause and cardiovascular-disease mortality: a cohort study of 80 306 British adults" is making rounds in all the media, e.g. this article in CBC interprets the findings as 

Cycling, swimming, doing aerobics and playing racquet sports offer
  life-saving benefits, a large British study suggests.

As far as I can tell the researchers did not have a control group, i.e. the didn't make a random group of people to do exercise and the other random group to not do exercise. How would they reliably address the selection bias? Maybe people who are less likely to die from a heart attack are exactly those who stick around and do tennis for 10 years? If your heart is weak, you'll keep feeling uncomfortable and quit the sport, or maybe there will be some other way through chemical released in the blood impacting your brain to quit the sport in this case.
They should take a large sample of people and force them do exercise regardless of their initial conditions. That's the only way to reliably understand the impact of sports in my opinion. The closest is probably a school, where everyone is supposed to take physical education classes. Even that is imperfect since kids can get all kinds of exemptions. 
 A: The title of the paper is "Associations of specific types of sports and exercise with all-cause and cardiovascular-disease mortality: a cohort study of 80 306 British adults"

Conclusions: These findings demonstrate that participation in specific
  sports may have significant benefits for public health. Future
  research should aim to further strengthen the sport-specific
  epidemiological evidence base and understanding of how to promote
  greater sports participation.

They don't claim causality.  They sort of imply it however.  They control for potential confounders, but probably can't achieve conditional independence.  
That said, I am no epidemiologist.  But in domain areas one can typically mix theory (i.e.: "we think that exercise is good") and weak evidence to make a stronger argument than could be made with weak evidence alone.
A: This question highlights two major issues. One is the inherent limitation of epidemiological research in terms of establishing causality. The second is the distortion of research results in the popular press.
The answer from @generic_user has covered the second issue. The authors did not claim causality. The dose-response relations for participation levels in some but not other types of exercise do suggest some underlying physiological basis, other than selection bias, for the results.
In terms of the limitation of epidemiological research, the types of controlled studies that you would like to see would be impractical or unethical even though they might better address issues of causality. Small-scale controlled studies, over limited periods of time, of some types of exercise have been done and some were cited in this research paper.
Under US standards, at least, you cannot "force them do exercise regardless of their initial conditions" unless they agree to be participants in a research study. This sets up a different type of selection bias.
The "control group" needed to compare against the forced-exercisers would presumably be required to abstain from exercise. Given the evidence from many types of studies and present knowledge of the physiologic effects of exercise, it would be unethical to prohibit people from exercising. 
Finally, the scale of the project would be daunting. From my own experience, it is difficult enough to oversee simple controlled observational studies of a few hundred participants over a few years. Tens of thousands of participants would be required, over the course of decades, to answer these questions in controlled prospective trials even if they were ethical.
A: It is perfectly possible to run clinical trials of exercise ethically. There must have been hundreds of them. In a typical such study a group of people would be screened for inclusion. They might be people with a pre-existing health condition, people with a risk factor for a health condition or just volunteers. They are then randomised into one of at least two arms. One arm will be treatment as usual (often called standard of care) where they receive the usual treatment for their condition. In the other arm(s) they would receive some sort of exercise.
The problem with the trials is that they are usually


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Small because there is no profit to be made from exercise so major players do not want to fund it


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Have short follow-up because of limited funds and also problems with loss to long term follow-up


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Often are difficult to generalise to the wider environment


Studies of the epidemiology of sport and other things like going for walks, dancing, gardening, and so are valuable as a complement because they show what happens in the real world.
