Question: Who owns meteorites?
Answer: Assuming the item in question is a meteorite and not space debris (which may belong to the equipment owner - e.g., government of the United States, Russia, China, France, IT capitalist, etc.), it belongs to the real property surface owner on whose property the meteorite is located.
Interestingly, this issue came up in regard to the largest meteorite ever found in the United States, the Willamette Meteorite. Ownership was settled by the Oregon Supreme Court in 1905. See Oregon Iron Co. v. Hughes, 47 Or. 313, 81 Pac. 572 (1905). Hint: It's not 'finders keepers' when, without the property owner's permission, you sneak onto someone else's private property and 'find' the meteorite that you then intend to 'keep.'
Question: If I find a meteorite on public land, may I keep it (or file a mining claim for the site)?
Answer: No. First, you cannot file a mining claims because meteorites are not a locatable mineral as defined by the mining law. Second, pursuant to the Antiquities Act, meteorites found on public land are considered objects of scientific interest and cannot be collected and taken for private use. This issue did in fact arise when individuals discovered the second largest meteorite ever found in the United States. The three ton meteorite - known as the Old Woman Meteorite - was found in 1975 on public land in the Old Woman Mountains located in San Bernardino County, California. Today, the Old Woman Meteorite is on display at the Desert Discovery Center in Barstow, California (actually, about 85% of the meteorite is on display there - a portion was removed for scientific analysis).