Of course, since the US Constitution was brought up, are there any restrictions or prohibitions on the Feds (or the states for that matter) on owning land?
The answer is: no.
I'm a little surprised that you would make this mistake, as ordinarily you're very well informed.
The powers of the Federal Government are restricted to only those delegated to it by the states and the people via the US Constitution. In other words, it's a government of strictly limited and enumerated powers. The Founders well understood this, but fearing that future generations might forget it, or be deceived, they insisted that one of the ten articles of the Bill of Rights should lay this out in clear language, and that took the form of the Tenth Amendment.
State governments are different, however, in that they were established as governments of plenary power, apart from specific powers denied to them by the US Constitution and by the various state Constitutions. In other words, the state governments have every power conceivable, except those prohibited to them, while the US Government (precisely the opposite) has no powers at all, except those delegated to it by the US Constitution.
Madison also makes this point clear in Federalist No. 45, which was written before the Bill of Rights.
"The powers delegated by the proposed Constitution to the federal government, are few and defined. Those which are to remain in the State governments are numerous and indefinite. The former will be exercised principally on external objects, as war, peace, negotiation, and foreign commerce; with which last the power of taxation will, for the most part, be connected. The powers reserved to the several States will extend to all the objects which, in the ordinary course of affairs, concern the lives, liberties, and properties of the people, and the internal order, improvement, and prosperity of the State."
