Read it a little closer. It refers to Territories and other land that the Fed can Constitutionally own. States are not Territories anymore and what the Feds can Constitutionally own within a State is spelled out in Art. 1 Section 8, Paragraph 17.
“To exercise exclusive legislation in all cases whatsoever, over such District (not exceeding ten miles square) as may, by cession of particular states, and the acceptance of Congress, become the seat of the government of the United States, and to exercise like authority over all places purchased by the consent of the legislature of the state in which the same shall be, for the erection of forts, magazines, arsenals, dockyards, and other needful buildings;”
So, the one you cited is limited to covering territories, “ten miles square” for DC and land “for the erection of forts, magazines, arsenals, dockyards, and other needful buildings.”
Not trying to cause a fight, just for your information.
Why doesn't Article IV, Section 3 make Article I, Section 8 meaningless?