"The federal government is Constitutionally allowed to own as much land as it wishes in any state with this caveat: The states must cede control to the federal government.
States and commonwealths that existed at the time the Constitution was written were given incentives to trade their western lands (think Ohio, Kentucky, Mississippi, etc.) to the federal government in exchange for debt relief and resettling native people. Texas entered the union with the provision that the state would control most of the land (which may be what Joshua's friend was thinking of.)
Following the Louisiana purchase, the federal government considered itself the sole owner of previously "unimproved" and unclaimed western lands that it purchased or otherwise acquired; so the states were de facto and de jure wholly owned by the federal government, which over time used various incentives to get people to move there . . . or not.
A somewhat interesting recap of federal laws concerning federal land ownership is
http://digital.library.unt.edu/govdocs/crs/permalink/meta-crs-1009:1"
The United States government has direct ownership of almost 650 million acres of land (2.63 million square kilometers) – nearly 30% of its total territory. These federal lands are used as military bases or testing grounds, nature parks and reserves and indian reservations, or are leased to the private sector for commercial exploitation (e.g. forestry, mining, agriculture). They are managed by different administrations, such as the Bureau of Land Management, the US Forest Service, the US Fish and Wildlife Service, the National Park Service, the Bureau of Indian Affairs, the US Department of Defense, the US Army Corps of Engineers, the US Bureau of Reclamation or the Tennessee Valley Authority.
This map details the percentage of state territory owned by the federal government. The top 10 list of states with the highest percentage of federally owned land looks like this:
Nevada 84.5%
Alaska 69.1%
Utah 57.4%
Oregon 53.1%
Idaho 50.2%
Arizona 48.1%
California 45.3%
Wyoming 42.3%
New Mexico 41.8%
Colorado 36.6%
Notable is that all these states are in the West (except Alaska, which strictly speaking is also a western state, albeit northwestern). Also notable is the contrast between the highest and the lowest percentages of federal land ownership. The US government owns a whopping 84.5% of Nevada, but only a puny 0.4% of Rhode Island and Connecticut. The lowest-percentage states are mainly in the East, but some are also in the Midwest and in the South:
Connecticut 0.4%
Rhode Island 0.4%
Iowa 0.8%
New York 0.8%
Maine 1.1%
Kansas 1.2%
Nebraska 1.4%
Alabama 1.6%
Ohio 1.7%
Illinois 1.8%
Even the 10th place is still below the two percent mark. One territory is not specified on the map: Washington D.C. It could be argued that this is the only main administrative division of US territory to be fully owned by the federal government. It could, but that would be wrong – and upsetting to those private citizens who own part of the nation’s capital in the form of their real estate. It would be more correct to state that the District of Columbia by default falls under the direct tutelage of the Federal Government."