If you had a bank account such as a checking or savings account closed at another bank due to overdraft charges that you owe then the bank would've most likely reported your account that you overdrawn on to the ChexSystems which tells other banks that you're a bad risk for a new checking or savings account.
The Report will stay on the ChexSystems database file for up to 5 years. After 5 years you should be able to open a new bank account with any other banks.
For now though you might look into an online bank account such as chimebank. These online bank accounts don't check ChexSystems or your credit for opening an account.