Is it possible to withdraw cash from an ATM without using a debit card?

Yes, cardless ATMs are now being deployed at many locations. You would still need to have some form of an account with the bank, just that the old-age need/necessity to swipe is no longer needed.

Paynet (in Kenya) is a great example of developing an ecosystem to for unbanked users, to have access to cash from an ATM by using cardless transaction.