It's unlikely if the non-government workers went on strike, the Capital would shut-down. Another reason DC doesn't have state-hood is to grant it a level of control and stability to perform it's function that can't be guaranteed by a State. When the Capitol was in Pennyslvania, there was a riot outside the state house that Pennyslvania failed to do anything about, despite having the authority and capability. That was a big reason for establishing a Capitol that had the independent authority to ensure it could protect its ability to continue to function, without having to rely on a state government. Yet another argument against state-hood.

In the implausible event all non-government service workers went on strike, DC would more than likely be able to continue to function.