A colloquialism referring to the city of Washington, D.C, and its role as the capital of the United States of America. The phrase is derived from Article I, Section 8 of the Constitution for the United States of America, which reads that Congress may force one or more states to cede a territory that is not larger than ten miles square (i.e. 100 square miles).
Modern-day Washington, D.C. city is less than 100 square miles because the part of the square that was taken from Virginia was returned. The appearance of D.C. itself plus Arlington County, Virginia suggests the original 100 square mile city with territory ceded by both Maryland and Virginia.
Meanwhile, back at Ten Miles Square, Federal politicians were proposing a different approach than one advocated by the Governors of many states.