No, because the District of Columbia is not a state (among other reasons).
Why would that have anything to do with it? It is still part of the United States of America and thus, still subject to federal laws, including the Bill of Rights.
But it has no militia. Therefore, there is no right to bear arms.
"A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed."
How does the part about the well-regulated militia have anything to do with the part after the comma, refering to the right to keep and bear arms?
"The people" refers to the militia. Back then, everyone was part of a militia.