"he didn't want to risk my getting infections from germs, scratches, bites, etc."
What in the world, you're going to encounter germs everywhere, at least with pets they give you comfort and love which is big help when you're going through something like cancer. I work at a specialty pharmacy and we have an oncology team, I've never heard a pharmacist mention "stay away from pets" or anything of the sort. That being said, it's possible that the patient just never brought it up. I would speak to the pharmacy you get the chemo medication from as to the effects of chemo (specific medication used) on pets (not a retail pharmacy as they likely deal with oncology meds a lot less), if your oncologist is the one who orders the medication and you're not sure where it's coming from just ask him or ask your insurance.
As to the chemo weakening your immune system and making you more susceptible to infection, people who take immunosuppressive medications are not encouraged to stay away from pets... It'll make the chance of an infection slightly higher over time, and make it harder to fight off infection but that doesn't mean you'll get an infection from your cats. That being said, if you do get cat bites from your cat during the time, I would consult your doctor to see if antibiotics should be recommended from the get go, just in case...
Sending healing thoughts your way, I don't know what type of cancer you have or what stage it's at and it's none of my business but hopefully the chemo will send you into remission quickly!