I was billed for a state E-file but then was told that I was unable to E-file for some reason. How can I get a refund?

The fee for using online TurboTax is the same whether you e-file or put it in an envelope and mail it.  The fee is not for e-filing; it is for using the software to prepare the return, so sorry, there is no refund for having to mail your state return.

Are you using the Desktop program?  There is a separate efile fee for state in the Desktop program.  You might be able to get that refunded.

 If you used the program installed on your own computer, and were charged an extra $19.99 or $24.99 for a state e-file, and you end up having to print and mail your tax return, you can contact customer support for a refund. 

If you filed with TurboTax online, the state charge is for the preparation module whether or not you e-file or print your return.