Putting money in the bank, and can hospitals be built with zakah funds?

Firstly:

Putting money in riba-based banks and taking riba for that, which is called interest, is a major sin, and paying zakah on that money does not absolve one of sin.

See the answer to question no. 22339, which explains the prohibition on riba.

The answer to question no. 181 explains the prohibition on putting money in riba-based banks.

Secondly:

With regard to what zakah may be spent on, it is not permissible to use it  for building a hospital or an orphanage, whether it is given hand to hand or via another means. The areas in which zakah may be spent are limited and it is not permissible to add anything to them. The areas in which zakah may be spent are mentioned in the verse in which Allah says (interpretation of the meaning):

“As-Sadaqaat (here it means Zakaah) are only for the Fuqaraa’ (poor), and Al-Masaakeen(the poor) and those employed to collect (the funds); and to attract the hearts of those who have been inclined (towards Islam); and to free the captives; and for those in debt; and for Allah’s Cause (i.e. for Mujaahidoon — those fighting in a holy battle), and for the wayfarer (a traveller who is cut off from everything)”

[al-Tawbah 9:60]

We have explained these further in our answer to question no. 6977

We have stated in a number of our answers that it is not permissible to use zakah money for building mosques or schools, or for printing copies of the Qur’aan. See questions no. 13734 and 21797.

But if the intention in giving the money to an orphanage is so that this money will be spent on the poor orphans, then this is permissible, if the orphans are poor.

It is better for you to dispose of the zakah on your wealth yourself, so that you can be certain that you have given it to one of the causes enjoined by Allah. You have to do your best to identify those who are entitled to it.

And Allah knows best.