What is the minimum food a person needs to eat on the Sabbath?
(I have an interest in water fasting. This means consuming nothing but water, for days or weeks on end. The power of the fast depends upon eating nothing at all.)
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Shulchan Aruch OC 288:2–3 says:
י"א שאדם שמזיק לו האכילה דאז עונג הוא לו שלא לאכול לא יאכל:
הגה וכן מי שיש לו עונג אם יבכה כדי שילך הצער מלבו מותר לבכות בשבת (אגור בשם שבולי הלקט):
Some say that a person who is harmed by eating, such that it is their delight not to eat, should not eat.
Note of the Rem"a: So too with one who finds delight in weeping, in order that their pain will leave their heart: such a one is permitted to weep on Shabbat. (Agor in the name of the Shibolei haLeket)
אדם המתענה בכל יום ואכילה בשבת צער הוא לו מפני שינוי וסת (פי' דבר קבוע) יש אומרים שראו כמה חסידים ואנשי מעשה שהתענו בשבת מטעם זה וכן אמרו שכך היה עושה ה"ר יהודה החסיד:
Regarding a person who fasts every day, and eating on Shabbat is painful for them because of the change in routine (explanation: a fixed habit), some say they have seen many pious and noble people who have fasted on Shabbat because of this reasoning, and they say that so acted Rabbi Yehuda he'Chassid.
So, an argument could be made that you may not have to eat anything at all. You would probably need to hear Kiddush from somebody, though, since you may not (eat nor) drink before Kiddush.
Still: It is preferable to eat something at each of the 3 meals. The Mishna Berurah in 291:1(2) cites an opinion that bread the size of a Kezayit - the size of an olive - may be sufficient in a pinch.
בכביצה - לאו דוקא אלא מעט יותר מכביצה דכביצה מקרי עדיין אכילת עראי כמ"ש סוף סימן רל"ב ולא חשיב סעודה [מ"א] וי"א שאפילו בכזית יוצא ידי הסעודה ונכון להחמיר לכתחלה אם אפשר לו:
Danny provided the Halachic (minimalistic) approach. This approach can be summarized as "חציו לכם": half of it for your own pleasure. However, there's an opposite Hassidic/Kabbalic/maximalistic approach worth mentioning.
Our tables hint at the main Altar ("זה השלחן אשר לפני ה'"), and our meals hint at the sacrifices. So, too, a man eats his meals on his tables mainly to please G-d (though he himself may enjoy the food too). Therefore, eating, especially on Shabbos can be seen as obligatory in some sort. That includes three things: bread (enough to bentch), meat, and wine. Since I follow this approach, I can force myself to eat, in order to represent the sacrifices on the altar.
When I was on a serious diet, eating less than 1000 Calories daily, I never changed my Shabbos eating habits, in deference to the Hassidic tradition.