How long might someone live If they give up on eating? Eating only 100-200 calories a day and a few small glasses of fluids
The currently recognized world record in fasting - by drinking only water and taking multivitamin supplements - is 382 days by a 27 years old man who went from 207 to 82 kg in this time (Postgradual Medical Journal, 1973, Wikipedia). This was a medically supervised experiment and should be considered a gross exception.
For a healthy adult with normal initial weight eating nothing or 100-200 calories worth of food per day, the realistic period to survive and remain healthy would be about a month (Scientific American, BBC).
I can't answer this for a cancer patient (one of the tags in the question is "cancer"). It is known that some people have attempted to cure cancer by fasting; this is a matter of a personal belief; scientifically it is not recognized as successful treatment. Fasting may reduce the side effects of chemotherapy, though (Aging, 2009).