I have read on another website that ʻAa'ishah, may Allaah be pleased with her, went to a public bath due to her illness. 1. What is a public bath? 2. Are women allowed to take a public bath, and why?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

We could not find any reference in the books at hand that the Mother of the Believers, ʻAa'ishah went to a public bath due to an illness or for any other reason, unless you are referring to the incident when she went out with Umm Mistah to relieve herself in the famous incident of Ifk (false allegation against Aaishah). You should know that she wanted to relieve herself and therefore went to an open space to answer the call of nature, which is equivalent to the present-day restroom. She did not go to a public bath in the modern sense of the word.

In general, public baths were bathhouses originally open for the public use, and they were also centers for healing. They were common in Muslim countries; some still exist in the present-day. It is impermissible for the Muslim woman to visit such public baths except in case of valid excuses such as washing from menstruation, post-partum bleeding, janaabah (major ritual impurity), sickness, or when she has any other need to bathe and cannot do it at home for fear of an illness or harm. Ibn ʻUmar reported that the Prophet, sallallaahu ʻalayhi wa sallam, said, “After some time, the lands of the non-Arabs will be conquered by you, and there you will find houses called public baths. So men should not enter them (to wash) except wearing their lower garments, and forbid the women to enter them except for a sick one or one in post-partum bleeding.” [Abu Daawood] Moreover, ʻAa'ishah narrated that the Prophet, sallallaahu ʻalayhi wa sallam, said, “Any woman who takes off her clothes anywhere but in her husband’s house has torn the veil that is between her and Allaah (i.e., He will expose her).” [Ahmad, Ibn Maajah, and Al-Haakim - Al-Albaani graded it saheeh (sound)] Umm Salamah reported that the Prophet, sallallaahu ʻalayhi wa sallam, said, “If a woman takes off her clothes in a place other than her husband's house, Allaah will tear His cover away from her (i.e., expose her).” [Ahmad, At-Tabraani and Al-Haakim]

Moreover, Muslim women are enjoined to keenly conceal their bodies; if the woman is in no need of visiting such public baths, it is disliked for her to visit them. Some of the Hanbalis even deemed it prohibited. Ibn Al-Jawzi and Ibn Taymiyyah asserted that if the woman is used to go to public baths and refraining from visiting them would constitute hardship, then it is permissible for her to visit them in case of an excuse. This means that it is permissible for the woman to go to the public bath, and she is obliged to conceal her ʻawrah (body parts that must be concealed as per the sharee'ah) therein. The majority of the scholars, except for the preponderant view of the Hanbalis, deemed it allowable for the woman to reveal of her body what is not considered ʻawrah before Muslim women. The ʻawrah of a woman in front of another woman is from the navel to the knee. Some scholars maintained that it is incumbent on the Muslim woman to cover her whole body in the public bath given the previously cited hadeeth narrated on the authority of ʻAa'ishah .

Ibn Qudaamah wrote:

“Chapter: Women are not allowed to visit public baths, given that they are enjoined to cover themselves, except in case of an excuse, such as bathing from menstruation, post-partum bleeding, sickness, or needing to bathe and they cannot do it at home because it is difficult for her or she fears illness or harm ... A concession is granted to women allowing them to go to the public baths in case of need as long as they lower their gaze and conceal their ʻawrah. But if there is no valid excuse, then it is not allowed. The Prophet, sallallaahu ʻalayhi wa sallam, said, ‘After some time the lands of the non-Arabs will be conquered for you, and you will find public baths, so forbid your women from them except for a menstruating woman or one in post-partum bleeding.' It has also been narrated that some women from Hims (Homs in Syria) entered upon ʻAa'ishah and that she said, ‘Are you the ones whose womenfolk go to the public baths? I heard the Messenger of Allaah, sallallaahu ʻalayhi wa sallam, say, ‘If a woman takes off her clothes anywhere other than in the house of her husband, then she has torn the veil that is between her and Allaah, the Almighty.’” [Al-Mughni]

Allaah knows best.