Devil in the details, as usual
I heard an in-depth discussion on the radio about this study last week. The author being interviewed was much less hysterical than the media coverage. Like any study, it's important to take a close look at where their data is coming from. Once you know that, a lot of the conclusions won't be surprising.
In this case, they were looking at successful suicides. The key finding is that suicide is often an impulsive act. If your method is slow-acting, it gives time to reconsider, or for others to intervene, making it less likely to succeed. If your method travels at about 1000 ft/sec, not so much. This is common sense. "'The lethality of the weapons drives the increased risk of suicide and homicide completion,' they wrote. 'Firearms have very high case fatality rates, particularly in the case of suicide. Guns leave very little room for reconsideration of the choice to end a life.'" In other words, people who attempt suicide with a gun succeed more often than those who try other methods. Voila, we arrive at the conclusion that just having a gun increases your risk of [successful] suicide. This is arguably true, although it ignores a lot of other relevant issues. It's a logical extension that anything that would slow a person down would give more opportunity for reconsideration or intervention; having guns locked up and unloaded would definitely slow the person down. They weren't advocating that weapons not be stored in the home, they were just pointing out the logical connection.
The conclusions of this study aren't newsworthy in and of themselves, but they're vague enough that the antis can take it and run with it, making all sorts of assertions well beyond what the study's authors may have intended. The headline of the linked article ("Gun ownership tied to three-fold increase in suicide risk") is a great example of this; it's a hyped-up oversimplification. It makes it sound like guns somehow radiate evil that convinces people to commit suicide, but that's not at all what the study reported.
Also from the article: "Anglemyer's team also found about a two-fold increased risk of death from murder among people who had access to a gun, compared to those without access to firearms. For women, the increased risk of being killed was even higher." (emphasis added) Ask yourself, what sort of women might disproportionately possess a gun? Answer: women in danger (from abusive exes or stalkers, living in high-crime areas, working high-risk jobs, etc.) Did the studies consider this? Probably not. Would it affect the results? Probably, but we can't know if it wasn't in the study. Catch-22.
Like so many studies, the danger isn't from the study itself, it's what ideologues and ignorant policymakers might do with it.