Success with running is based heavily on the strength to weight ratio of your legs, that is how strong are your legs and how much do you weigh?  From what you said, your fitness level appears low, and you’re trying to change that. 
The problem is that a thin and strong runner has strong legs and low body weight.  When the trained runner runs, the percentage of max-effort in their legs is very low, and to them it is truly an endurance activity.  They can figuratively run forever with minimal to know discomfort. 

The reason is that there are multiple metabolic pathways your body uses under different circumstances, and there are adaptation to being both active and inactive. Metabolically, the fit and thin runner on a long-distance slow-pace training day is using oxygen exclusively, and is slowly metabolizing fat. The runner has efficient breathing, a low heart rate, and using oxygen for metabolism doesn’t produce very many nasty waste products. The runner has also developed a very dense network of blood vessels that squirm all throughout their leg muscles, to ensure oxygen gets in and waste gets out. Being non-fit, and having a low strength-to-weight ratio, is a much different situation. First, your muscles are working at near 100% max capacity, and at this work load, the body stops using oxygen for metabolism and switches over to non-oxygen pathways for immediate energy. The side effects of these non-oxygen pathways are nausea, and if trained hard enough, you will vomit, or at least wretch for a while.
Your lung efficiency is probably low (which will change, as you practice using your lungs), and I’d assume your heart rate is reaching maximum during these bouts of nausea and effort. Your blood vessel (capillary bed) density is also probably very low, preventing oxygen from getting to your muscles, and preventing waste from leaving.
When I teach running to newbs, I teach 5x5 squatting at the same time, as strength-to-weight ratio is absolutely critical to running. I highly recommend you find a knowledgeable strength coach to help you start squatting. I also highly recommend getting a heart rate monitor, as your heart beat can tell approximately where you are metabolically, and you can avoid the higher effort aspect of running that is making you feel ill. Keep with it though. The body will adapt to whatever stimulus you expose it to, guaranteed.