It really depends on what the illness is. A lot of medical problems will disqualify you from military service. Luckily, most of them are waiverable, but some are not. Any previous illnesses or surgeries will need to be reported to DODMERB (Department of Defense Medical Evaluation Review Board). You will need to submit any past medical records as well as any new information/letters from a provider that says you are now fit for duty. This will be reviewed, and you may be asked to see another provider for another exam. If all goes well, a waiver will be approved, and you will be allowed to join. Do not lie about your medical history! If they find out, you will be charged with fraudulently enlisting, which is a felony, and will be dishonorably discharged from the military. It is not worth it, so just wait for the waiver process. As for being overweight, the following chart is used:
Air Force applicants who exceed weight requirements shown in the below chart are measured for body fat. Body fat standards are no more than 20% for males under age 30, 24% for males age 30 or over, 28% for females under age 30, and 32% for females age 30 or over. If you don't meet the weight requirements or the body fat requirements, you cannot join. I do not believe this is waiverable.
It's hard to answer your question generally. If you had surgeries/illnesses that left you debilitated for life in some way, then yes those could preclude you from joining. As far as the weight issues goes, if you can get within standards you're Ok. If you're morbidly obese and there's no chance of that, then you'll have a problem. Unless you want to go into more detail here, talk to your civilian doctor, get copies of your medical record, then go talk to a recruiter and voice your concerns. (The recruiter should be honest with you, there's no sense in doing the paperwork on someone who'll fail a screening physical.)
A good way to get a solid list of what they want it to go to the USUHS website and check out the catalog. They have a whole appendix about physical requirements and such.
You can easily look up the health criteria for military service by doing a google search.
But you can get a waiver for almost anything. Especially if you are going to be a physician, dentist, or vet.
From my own experience, the Army is more liberal with waivers than the Air Force. But after learning more about the military and military medicine, the Army is much, much better than the Air Force anyway.