One Israelite is forbidden to lend to another Israelite at interest. Is it even possible to work in the modern banking industry without sin? I don't see how.

There are 3 tiers in regards to lending at interest:
-It's ok to charge interest to unbelievers/foreigners.
-It's wrong to charge interest to your brothers.
-It's worse to charge interest to poor brothers.
#exodus 22:25

An interest-free loan is usually better than charity because it preserves the borrower's self respect. Even so, lend without expecting repayment.