Yoma 12A

והא ⁠— דכרכים, והא ⁠— דכפרים.

and this baraita, which states that a synagogue does not become impure, is referring to synagogues in large cities. Since those synagogues attract people from different places, the building is not the property of the local residents but that of the public. And that baraita, which states that a synagogue becomes impure, is referring to synagogues in villages, which belong solely to the residents of the village, and its status is like that of a house owned by partners.

ודכרכים אין מטמא בנגעים? והתניא: ״אחוזתכם״, אחוזתכם מטמאה בנגעים, ואין ירושלים מטמאה בנגעים. אמר רבי יהודה: אני לא שמעתי אלא מקום מקדש בלבד. הא בתי כנסיות ובתי מדרשות מטמאין בנגעים, ואף על גב דכרכים נינהו! אימא: אמר רבי יהודה: אני לא שמעתי אלא מקום מקודש בלבד.

The Talmud asks: And do the synagogues in large cities not become ritually impure with the impurity of tzara'at? Wasn’t it taught in a baraita that it is written: “In a house of the land of your possession” (Leviticus 14:34); the land of your possession becomes ritually impure with the impurity of tzara'at, and the city of Jerusalem does not become ritually impure with the impurity of tzara'at, since it belongs to all the Jewish people rather than to a specific tribe? R' Yehuda said: I heard that it is only the site of the Temple [mikdash] alone that does not become ritually impure with the impurity of tzara'at. It can be inferred that in the opinion of R' Yehuda, even synagogues and study halls in Jerusalem become ritually impure with the impurity of tzara'at, and that is the case even though they are synagogues in large cities. The Talmud rejects this; rather, one must emend the baraita and say that R' Yehuda said: I heard that it is only a sacred [mekudash] site alone. That definition includes synagogues and study halls.