What is the minimum food a person needs to eat on the Sabbath?
(I have an interest in water fasting, which means not consuming anything but water for days or weeks. The power of the fast depends on not eating anything at all.)
Like any library, Mi Yodeya offers tons of great information, but does not offer personalized, professional advice, and does not take the place of seeking such advice from your rabbi.
The Shulchan Aruch in O"C 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: And so 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 heChasid.
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.
Though it is preferable to eat something at each of the 3 meals. The Mishna Berurah in 291:1(2) brings 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) which can be summarized in "חציו לכם" (half of it for your own pleasure) while there's an opposite Hassidic/Kabbalic/maximalistic approach worth to be mentioned.
Just as our tables hint on the main altar ("זה השלחן אשר לפני ה'") and our meals to the sacrifices, a man eats his meals on his tables to please G-d, not for his own pleasure (nobody said he can't enjoy it though). Therefore eating especially on Shabbos can be seen as obligatory in some sort and that includes three things: bread (enough to bentch), meat, and wine. So a person (that follows this approach, myself included) can force himself to eat, just to represent the sacrifices on the altar.
When I was on a serious diet eating under 1000 calories a day, I never changed my Shabbos habits in my respect to the Hassidic tradition.