Is it permissible for him to live with his paternal uncle who has adolescent daughters?
What we advise you to do is not to live with your uncle and his family, because you are not a mahram for your uncle’s wife or for his daughters, and this will cause you and them a great deal of restrictions if you live with them.
The Prophet (peace and blessings of Allaah be upon him) forbade entering upon women, and when the Prophet (peace and blessings of Allaah be upon him) was asked about the husband’s relatives, he spoke more sternly about them than about anyone else. So listen to the hadeeth and read what the scholars have said about it.
It was narrated from ‘Uqbah ibn ‘Aamir that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Beware of entering upon women.” A man from among the Ansaar said: O Messenger of Allaah, what about the in-law? He said: “The in-law is death.”
Narrated by al-Bukhaari, 4934; Muslim, 2172.
Al-Nawawi (may Allaah have mercy on him) said:
The scholars of Arabic language are agreed that the word ahma’ (in-laws) refers to the relatives of a woman’s husband, such as his father, brother, nephew, cousin and so on.
With regard to the words of the Prophet (peace and blessings of Allaah be upon him), “the in-law is death”, what this means is that the fear with regard to him is greater than the fear with regard to others, and evil is expected from him and the fitnah is greater because he is able to reach the woman and be alone with her without being denounced for that, unlike one who is a stranger.
What is meant by the in-law here is the relatives of the husband, apart from his father (and grandfather) or his sons. Fathers and sons are mahrams for his wife and it is permissible for them to be alone with her, and they are not described as being death. Rather what is meant here is the brother, nephew, uncle, cousin and so on, who are not mahrams. The custom among people is to take this matter lightly, so a man may be alone with his brother’s wife. This is what is meant by death, and he should be prevented more than a stranger for the reasons we have mentioned. What I have mentioned is the correct meaning of the hadeeth.
Ibn al-A’raabi said: This is a phrase that is used by the Arabs, as it is said: The lion is death, i.e., meeting a lion is like death. Al-Qaadi said: what is meant is being alone with the in-laws (the husband’s relatives) which may lead to fitnah and disaster for one's religious commitment, so it is likened to the disaster of death in order to demonstrate the seriousness of the matter. End quote.
Sharh Muslim, 14/154
You can see that this hadeeth applies to your situation, if you enter upon your uncle’s wife or daughters who are non-mahrams for you, so what about the ruling if you live with them?
Secondly:
If you have no alternative but to live with them, or you are going to stay there temporarily until you find another house, then you must pay attention to the following matters:
1 – Avoid being alone with your uncle’s wife or one of his daughters.
It was narrated from Ibn ‘Abbaas (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “No man should be alone with a woman and she should not travel unless she is accompanied by a mahram.” Narrated by al-Bukhaari, 2844; Muslim, 1341.
2 – Both you and the women should lower the gaze and avoid looking at one another.
Allaah, may He be exalted, says (interpretation of the meaning):
“Tell the believing men to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts). That is purer for them. Verily, Allaah is All‑Aware of what they do.
31. And tell the believing women to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts) and not to show off their adornment except only that which is apparent (like both eyes for necessity to see the way, or outer palms of hands or one eye or dress like veil, gloves, headcover, apron), and to draw their veils all over Juyûbihinna (i.e. their bodies, faces, necks and bosoms) and not to reveal their adornment except to their husbands, or their fathers, or their husband’s fathers, or their sons, or their husband’s sons…”
[al-Noor 24:30-31]
3 – When you speak to them or they speak to you, there should be no softening of the voice or speaking in an alluring tone.
Allaah, may He be exalted, says (interpretation of the meaning):
“O wives of the Prophet! You are not like any other women. If you keep your duty (to Allaah), then be not soft in speech, lest he in whose heart is a disease (of hypocrisy, or evil desire for adultery) should be moved with desire, but speak in an honourable manner”
[al-Ahzaab 33:32]
4 – Your uncle’s wife and daughters should observe proper hijab in front of you, covering the entire body.
It is better for you not to live with them, and to look for a house of your own, so as to relieve them and you of any shar’i restrictions, and so that you will not be a cause of hardship and restrictions for them. If their house is big and you can have a separate room with all amenities, then it is permissible for you to live there, but as for living in a room in their house, and sharing the facilities of the house, we do not think that that is permissible, and we think that the conditions that we have mentioned for it to be permissible will be difficult for many people to meet.
See also the answer to question no. 13261 for important information.
And Allaah knows best.