If you eat a hamburger from McDonald's you will most likely get sick. If you eat fresh, wild caught Alaskan salmon, you will not. Eating meat is not bad for you. 20th century Eskimos (before the introduction of high-carb western diets and smoking) had virtually no cancer or heart related diseases within their population, and their diet consists of 99% meat. Obviously, meat is not unhealthy, it's just a matter of being informed. As a rule of thumb:
1-2 servings of red meat a week. Get grass fed beef or wild bison if possible.
4-5 servings of fish a week. Small wild caught fish is extremely low in mercury. Like sardines and salmon for example. Large, predatory fish, like sword fish, even wild caught will have at least moderate levels of mercury so it's better to eat lower on the food chain.