What is the ruling on purchasing an apartment with the help of a bank, involving interest, in order to escape shared accommodation in a non-Islamic country, for the sake of preserving dignity, religion, and privacy?

Your question comprised two matters:

First: Your living with non-mahrams in one house, sharing facilities. This is not permissible if it leads to unlawful khalwa (definition) (seclusion of a man and a woman who are not close relatives), unlawful ikhtilat (free mixing of unrelated men and women), or non-adherence to hijab (definition), or similar forbidden matters according to Sharia (definition). It is permissible if it does not lead to such forbidden matters, provided Islamic manners are observed. However, it is better to live separately from them.

Second: Taking out a riba-based loan from a bank to buy a house. This is unlawful because it is not permissible to engage in a usurious loan except in cases of dire necessity. What you mentioned is not among these, as it is possible to rent another house, or move to another country, or live with Muslims who adhere to Islamic regulations.