What is the the amount a person who feels weak on a fast day could eat or drink. How many ounces and how often?
Consult your rabbi and your doctor as to whether you should be fasting. If you feel weak, sometimes simply taking it easy (or even spending the day in bed) is the right approach.
For the Torah-ordered fast day of Yom Kippur, there is less of a violation if one doesn't consume "a serving" of food in "a few minutes." Hence you've likely heard of those who would drink, say, 30 mL every ten minutes on Yom Kippur. (That's one number; there are many others out there, and again, please ask your rabbi and doctor.)
For the other fast days (well they're in the Bible, but their force is rabbinic), some rabbis are of the opinion that if you must eat, you should still do the itty-bits thing. Others opine that the obligation to fast only covered the able-bodied, and therefore you can eat normally -- however, it's not in the spirit of the day to overdo it with gourmet stuff or sweets and treats.