They can either not get any illnesses at all, or they can get one.
I've played SA2 through 100% twice (my save was deleted once :P), and none of my chaos ever got sick as far as I can remember. The doctor always seems to say your chao is fine, even if you'd beat it up and throw it around all the time.
Chat with our AI personalities
You have to buy it
Sonic Advance 1, 2 and 3 (Gameboy Advance) have chao gardens which are small and not very great, but Sonic Adventure (Dreamcast), Sonic Adventure DX (GameCube), Sonic Adventure 2 (Dreamcast), Sonic Adventure 2 Battle (GameCube) all have chao gardens where you can do the most things and raise your chao the best and most effective. Sonic Adventure 2 Battle is the most convinient and shows you all you need to know about a chao, and it's a really good game to raise them in by using animals (and Chaos Drives) as a means of training them. Sonic Advance 1 and 2 can connect up to Sonic Adventure 2 Battle and Sonic Adventure DX via a Gameboy to GameCube link cable and you can exchange chao with.
buy it in the black market
When ignored or abused I believe
Attack it. Also, it's Sonic Adventure 2 Battle.