The Washington Post put up an interactive map of the top 100 metro areas with federal workers. The Seattle-Tacoma area has 86,000 federal workers representing just under 5% of the total employment. In the graphic, the bigger and darker the circle the bigger the percentage of the local work force.
This is not directly reflective of federal shutdown furloughs or other associated job losses as it only uses the rough numbers of federal employees in a given area and only shows the top 100 locations. Smaller population areas might be even more impacted. Joint Base Lewis-McCord will definitely see furloughs, but of course many of those employees are considered essential. But on the flip side, the joint base also has a great many non federal employee contractors. The map leaves off other significant military employment centers in Washington State - namely Oak Harbor with the naval air station, Kitsap County with the Bremerton ship yards and Bangor submarine base, Everett and its naval ship yard and Spokane with the Air Force station. Then there are the smaller communities with forest service offices - it doe not take many employees in Quilcene to have a impact on the local economy.
One of the largest areas of federal spending in Washington State has been the Tri-City area via Hanford projects. The vast majority of that work is large federal contracts. Cuts in funding have a large impact on the economy of the Tri-Cities. Stimulus spending actually caused the Tri-Cities to add jobs during the large down turn in the economy and as the stimulus spending ended the area lost jobs.