In this source sheet, we're going to address two questions.
1) Can one feed a baby in synagogue, regardless of how one does so?
2) Can one breastfeed in synagogue/in public?
The answer to both is yes. The purpose of this sheet is to provide the sources that back up that claim.
Section 1:
Can one feed a baby in synagogue?
בתי כנסיות ובתי מדרשות אין נוהגין בהן קלות ראש כגון שחוק והיתול ושיחה בטלה ואין אוכלין ושותין [...] ותלמידי חכמים מותרין לאכול ולשתות בהן
In synagogues and houses of study, one does not act in an inappropriately thoughtless fashion, such as with joking, and comical behavior, and pointless speech and one does not eat and drink [...] but Talmidei Chachamim are allowed to eat and drink in them.
(5) And they are allowed - And this is the rule as well, that if the people of the city need - occasionally - to feed poor guests or provide them somewhere to stay, they may also eat in the synagogue. But this is just for synagogues outside of Israel and some forbid even there, but it seems that, when the situations demands it, one should not be strict.
בהם מדוחק. נ"ל דאורחים שרי לאכול שם דהא איתא בסימן רס"ט דאורחים אוכלים בבי כנשתא וה"ל קידוש במקום סעודה וכן הביא ב"י בסימן זה בשם רמב"ן:
When absolutely necessary. It seems to me that guests are (also) allowed to eat in synagogue, like it says in Siman 269, that guests eat in the synagogue and they say kiddush in the place where they will eat, and the Beit Yosef quotes this opinion there in the name of the Ramban.
Bottom line: The prohibition against eating in synagogue is clearly a matter of respecting the sanctity of the place and is set aside whenever there is some kind of counter-pressure. Even if such concerns apply to babies, and there is good reason to think that a baby's need to eat puts them in a more lenient category, feeding a baby is clearly enough of a counter-pressure, especially when one takes into account how willing contemporary synagogues are to allow young children to eat (and throw) candy within the sanctuary.
Section 2:
Can one breastfeed in public/in synagogue?
To ask the question differently: is there any problem with nursing in public or in synagogue, even without a cover, because one's breasts are inevitably exposed for a period of time?
הָאִשָּׁה יוֹשֶׁבֶת וְקוֹצָה חַלָּתָהּ עֲרֻמָּה, מִפְּנֵי שֶׁהִיא יְכוֹלָה לְכַסּוֹת עַצְמָהּ, אֲבָל לֹא הָאִישׁ.
A woman may sit and separate hallah while she is naked, since she can cover herself but a man may not.
By "cover herself", the Mishnah means that she can close her legs and thus hide her genitalia. We infer two things from this Mishnah. First, that what constitutes nakedness in a woman is her genitalia, as there is no need for her to cover her breasts. And second, that she can both perform the commandment and make the blessing while her breasts are visible.
The following two stories from the Talmud also support the idea that, as far as the rabbis were concerned, breasts are not ervah, nakedness.
תנו רבנן דרש רבי יוסי הגלילי בשעה שעלו ישראל מן הים נתנו עיניהם לומר שירה וכיצד אמרו שירה עולל מוטל על ברכי אמו ותינוק יונק משדי אמו כיון שראו את השכינה עולל הגביה צוארו ותינוק שמט דד מפיו ואמרו זה אלי ואנוהו שנאמר (תהלים ח, ג) מפי עוללים ויונקים יסדת עוז
§ The Sages taught in a baraita that Rabbi Yosei HaGelili taught: At the time that the Jewish people ascended from the sea they resolved to sing a song of gratitude to God. And how did they recite this song? If a baby was lying on his mother’s lap or an infant was nursing from his mother’s breasts, once they saw the Divine Presence, the baby straightened his neck and the infant dropped the breast from his mouth, and they recited: “This is my God and I will glorify Him” (Exodus 15:2). As it is stated: “Out of the mouths of babies and sucklings You have founded strength” (Psalms 8:3).
So what changed?
The sexualization of the breast, already under way by the 19th century, was accelerated by the World War II pinup girl poster, postwar soft porn such as Playboy magazine, and the popularity of such Hollywood icons as Marilyn Monroe. The result created an incongruity of the breast as a source of infant nutrition. As breasts became more sexualized, they became less functional: more the purview of men as sexual objects and less the domain of infants and as a source of food. As this transformation continued, breast-feeding, especially in public, became less normal and more taboo, and by midcentury most Americans attached a vague sense of disgust to the practice. Now that breasts were primarily sexual the idea of women breast-feeding infants, especially in public but even in private, felt abnormal and destabilizing. Modernity apparently did not include breast-feeding women; by implication breasts were for men and sex.
By contrast, societies with strong breast-feeding practices tended to be developing countries, many populated with nonwhite majorities. Such countries, not surprisingly, were apt to be more tolerant of exposed breasts and breast-feeding in public, one of many factors that caused some Americans to deem them less civilized in comparison with the United States and other developed nations. To most Americans the idea of exposed breasts and suckling children elicited too much discomfort, was too reminiscent of the dark-skinned women from developing countries displayed in full color on the pages of National Geographic. “Proper breastfeeding and care of the baby is essentially a primitive activity … far removed from modern practices,” observed the medical philosopher Simon S. Levin; in other words, our modern, technologically advanced society has rendered breast-feeding obsolete.
The Ben Ish Hai (Hakham Yosef Hayyim) lived from 1835 - 1911 and provides a more modern perspective on the status of breasts in Halakha.
He differentiates between actual nakedness and areas of the body that are habitually covered. While the latter are not nakedness in the Halakhic sense, they still have a certain sense of privacy about them and one is supposed to wash one's hands if one touches a body part that usually remains covered.
ואלו דברים שאין צריכין נטילה כי אם רחיצה כל דהו, דהיינו שיטול ידיו פעם אחת משום נקיות, ואלו הן:
[...]
וכן הדדין בזמן שמנקת את בנה שדרכן להיות מגולין תמיד אם נגעה בהם אין צריכה רחיצה, אבל אם אינה מנקת שדרכן להיות מכסות אותם לגמרי אז הם בכלל מקומות המכוסים, וצריכה רחיצה אם נגעה בהם...
And these are the things that do not require one to ritually wash one's hands (Netilat Yadayim), but merely require regular hand washing for cleanliness reasons, and these are them:
[...]
And so too the breasts when a woman is nursing her child, because that is a time when they are constantly exposed. If she touches them, she does not even need to wash her hands. But if she is not nursing, which is a time when the breasts are covered, then they are just like all other covered places and she should wash her hands after touching them.
The Ben Ish Hai picks up on something that many women instinctively feel when they are nursing - that there is a qualitative difference between her breasts when she is nursing and at other times. They are less private in every sense: they are more often seen, they belong to someone else who needs them to eat, and having them exposed doesn't bother her the way it might have before.
The law recognizes the experiential distinction and reflects it.
Bonus Section 3:
Praying while holding a child. While most of the sources think this is fine, the Mishnah Berurah cites a law that you cannot place a child in front of someone while they are praying.
I would humbly suggest that this means that, while I can begin praying while holding my child, if my partner takes the child from me, it's on them to entertain the baby until I am done with the Amidah.
And for much more information and detail, see here:
https://library.yctorah.org/lindenbaum/breastfeeding-and-showing-affection-in-shul/