The thought that "people aren't supposed to live here" makes no difference to the matter; people do live there, a lot of people, more than the entire states of Wyoming and Vermont, and without those residents the government would cease to function.

It would be feasible to enfranchise the citizens of DC by granting statehood while also maintaining the original intent of the city, by requiring the new state constitution to yield all physical security to the authority of the federal government.

Then return the parts of the current district on which residences (other than official government ones like the White House) sit to the original states.

Because people do, in fact, live in D.C., and those people are currently governed (and taxed) without representation in the federal legislature.

[1] As noted below, a significant fraction of the African American population live in D.C. because their ancestors were brought here involuntarily. They should be permitted to vote.

That overstates the matter. The federal government was incapable of defending the District militarily against the states, because it intentionally had no significant standing army at the time the District was ceded to it [0]. Even after the U.S. Army started to take shape — mainly for service on the frontier, not at the national capital — it's doubtful that the District could have resisted invasion by the militia of even one of its state neighbors; this is indirectly evidenced by the ease with which the British took and burned Washington in 1814.

[0] https://en.wikipedia.org/wiki/History_of_the_United_States_A...

Although it runs into the problem that Virginia and Maryland don't want it.

The only people with a rational (short-term, at least) interest in D.C. representation is D.C. residents, and they don't get a vote, or even a vote for people who get a vote.

This strikes me as a very truthy claim. I'd expect the vast majority of people to be utterly indifferent to paying Virginia or Maryland to absorb parts of DC. They might say otherwise in a poll, but they'd act like it never happened when they next voted.

DC statehood would also help mitigate the distortive effect of allocating 2 Senators to every state regardless of population, which exaggerates the rural vote.