The answer is, it depends. Typically you would be able to open a company bank account, where at least one representative from your company has to reside inside the United States and be given the authority to sign on the account and be able to conduct banking transactions. This is (to the best of my knowledge) the only legal way of doing it.
You cannot open a personal bank account without physically being present in the United States. You might be able to open an HNI (High Network Individual) bank account, but to truly activate it, you would still need to come and visit a physical branch in the United States.
As far as a US Bank Account and Routing Number is concerned (a DDA – Direct Deposit Account), you can open one in one of the US Territories, like Puerto Rico. This is the closest thing you will get to a US Bank Account.