Nisaa
The majority of scholars, including the four Imams, hold that a person in Janabah may not enter a mosque and stay there. Passing straight through without stopping is a different matter, and is agreed to be permitted, particularly when there is a genuine need to cross through it rather than go around.
Their primary evidence is the ayah (interpretation of the meaning): “O you who believe, do not approach prayer while you are intoxicated until you know what you are saying, nor while in a state of Janabah, except as one passing through, until you have washed.” [An-Nisa' 4:43]. This is the same ayah cited for a menstruating woman and the mosque, where hayd is reasoned to it by analogy (Qiyas). For Janabah itself the ayah applies directly rather than by analogy, since it names Janabah by name.
Commentators have read the exception, “except as one passing through,” in two different ways. Ibn Mas'ud and Anas ibn Malik (Radi Allahu Anhum), along with al-Hasan al-Basri and Ibrahim al-Nakha'i, took it at face value: a person in Janabah may walk through a mosque without stopping. 'Ali and Ibn Abbas (Radi Allahu Anhum), along with Sa'id ibn Jubayr, read it instead as addressing a traveller who is in Janabah and cannot find water, permitting him to pray with Tayammum rather than wait until he can make Ghusl. The ruling followed by the four madhabs, forbidden to stay, permitted to pass through, rests on the first reading. The second remains a real, attested position on the same ayah, not a ruling about entering a mosque at all.
A hadith of Aishah (Radi Allahu Anha) is cited alongside it. The Prophet (Salla Allahu 'Alayhi Wa Sallam) instructed the Companions to turn the houses that opened directly into the mosque away from it, and said: “I do not permit the mosque to a menstruating woman or to one in Janabah.” (Narrated by Abu Dawud). Its authenticity is genuinely disputed: al-Bukhari, reviewing its chain in al-Tarikh al-Kabir, judged that it is not established as authentic from the Prophet, while other hadith scholars have accepted it as supporting evidence for the ruling above. Where it is cited, it stands alongside the ayah rather than as a proof on its own.
The Hanbali school records a concession the ayah does not spell out on its own. Ibn Qudamah notes in al-Mughni that a person in Janabah who needs to enter the mosque, to give the call to prayer or to retrieve something left inside, for example, should perform Wudu first and then enter. Wudu does not lift Janabah, and Ghusl is still owed before she can pray, but it is treated as enough to permit the entry itself when there is an actual need for it.
None of this changes what causes Janabah in the first place, or when Ghusl becomes obligatory. The same state keeps the Qur'an out of reach until Ghusl is made too, both reciting it and touching the Mushaf, which follows the same restriction and the same barrier exceptions as hayd.