No, when you receive unemployment from a state, they report that income to the IRS on a 1099-G form and you are required to include those numbers in your return. TurboTax Online Free does not include that form, but it is supported in TurboTax Online Deluxe and the other versions.
You can begin using TurboTax Online Deluxe for free and do not have to pay until you are ready to file. If you have a federal refund due, you may qualify to have the TurboTax fees paid out of the refund.
Here is a link to a product page which has additional information about the Deluxe version. TurboTax Online Deluxe