Well im guessing you're talking about what could happen if you overexert yourself. Well if you're new to training, then you have to make sure to pace yourself and keep yourself hydrated. Don't try to match what the person next to you is doing or compete with the guy who's been working out for 20 years. You'll end up getting dizzy, nauseous or even physically hurting yourself such as pulling a muscle or tearing something. Just gradually build yourself up.
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you don't catch illnesses from fitness activities themselves. you might catch something if you use equipment right after someone that may have been sick if you don't clean it.
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The only real illness that can occur other than malnutrition from simply not eating enough, would be ketone-acidocis. It happens when you lose too much fat too quickly, and toxins from the breakdown of fat build up in your body. Just make sure you drink plenty of water and keep your electrolytes up and you have nothing to worry about.
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You do not incur any illness as a result of exercising!!

Some illnesses such as thyroid problems or asthma may make it a little harder to exercise but that's another issue.

If you don't know what you're doing you may injure yourself, but again that's another issue as an injury is not an illness.

Common exercise related injuries are people using free weigts incorrectly and ending up with muscle strain, or running in the wrong shoes and ending up with ankle or knee problems etc.
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Personal Trainer