The Federal government assumed control of most of the land in the west as previously unowned in territories not yet divided into states. It didn't have to pay for it, over time it simply reversed the process of recognizing settlement of unowned land and seized it for itself as permanent control.

Private land that is acquired, including by eminent domain, by the Federal government (and much state acquisition) for parks and wilderness is funded mostly through the Land and Water Conservation Fund (LWCF) from the Great Society programs of the 1960s. That law authorized the funding and Congress annually appropriates the money up to $900 million/year.

The anti-private property preservationist lobby (Audubon, Wilderness Society, National Parks Conservation Association, Sierra Club, etc. etc.) is trying to turn that into a permanent $1 billion (or more) off budget annual "trust fund" entitlement for acquisition bypassing Congressional appropriations. They also want to expand it into an entitlement slush fund for activist groups and all levels of government for land use and acquisition planning. Paul Ryan's omnibus bill just extended LWCF instead of killing it.

The maintenance of Federal lands is also paid by Congressional appropriations in the general budget, i.e., taxes. The National Park Service is billions in arrears for maintenance and rehabilitation but wants to keep acquiring more in an insatiable mission mentality of expansionism.

Agreed. However with Agenda 21, more that likely the U.S. government will give the land to the United Nations for "preservation" meaning no one can enter, use, or enjoy as it is a "human-free" area. If the lackey in washington thought that would gain them anything; it would have already been done.