I'm pretty sure you would be eligible. I think the maximum DC benefit is around $370 per week and is subject to taxes, although you can choose to have them taken out now or pay them later when you file 2013 taxes.
I am a contractor worried about losing my job (at least temporary furlough) during the shut down. Would I be eligible for unemployment benefits in DC?
Do you know what they are and if they are subject to taxes?