Is it permissible to take a loan from the bank to help my sister and her husband in their financial distress and to provide capital for a new project?

May Allah reward you for your good intention to help your sister and her husband, for the Prophet, peace and blessings be upon him, said: "The most beloved of people to Allah are those who are most beneficial to people." If the loan you intend to take out is for a legitimate financial transaction with an Islamic bank, then there is no harm in it, and you will be rewarded for your good intention. However, if the intention is to borrow with usury (riba), then it is not permissible, and what you mentioned does not make it permissible for you to proceed. You can help your sister through other permissible means.