At the risk of stepping into the middle of a fight… I must suggest that anyone interested in the relationship between fatness, fitness, and health read a fantastic, complete, and highly accessible scientific review on this subject written by Vojtech Hainer, Hermann Toplak, MD, PHD, and Vladimír Stich, MD, PHD. For the TL;DR crowd:

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1. It is inarguable that being fat and sedentary is terrible for your health.

2. Quite a few recent, large, and well-crafted studies have dug into figuring out which is more important for preserving health: being normal weight or being active. Almost all—and particularly the better crafted studies that have used independent and scientifically rigorous measures of fitness rather than surveys—show being active is the more important factor, even if the activity does not succeed in causing one to lose weight.

The relative risk of dying per year, by fitness and fatness.
  • The relative risk of dying per year, by fitness and fatness.

The numbers in this figure represent the relative risk of dying in a given year, broken down by fitness and fatness; lower numbers are better. The healthiest group in this study were the (moderately) fat, fit group.

A prospective JAMA study from 1995 showed nicely that becoming fit had a significant effect on both all-cause mortality and cardiovascular mortality. The effect size wasn't small. For every minute one could last longer on a treadmill, the relative risk of death in a given year was reduced by 7.9 percent.

3. Therefore, just because someone is fat (or looks fat) doesn't mean they're unhealthy. Additionally, even if exercise didn't work for you (i.e., it didn't make you thin), you are still deriving a massive benefit from exercising even if you remain fat. The amount of exercise needed is quite small, 20 minutes of moderate exercise (enough to get your breathing rate up, not to drive you into a drenching sweat and gasps for air), three times a week is enough.

Updated:
Here is data for women, around the same question. This study isn't quite as clean as the one for men I cited above.

Relative_Risk_women.png

In this study—the Lipid Research Clinics Study—both fatness and fitness were relevant predictors for health. A more ethnically diverse population of subjects was involved in this study. The results from slightly overweight and massively overweight individuals were combined, perhaps clouding the relationship in the other (men-only) study was able to demonstrate.