The US government just shut down, stopping funding for (and thus closing) most National Parks, BLM lands, campgrounds on national lands, etc...
What areas are still open?
Essentially, any places which has Federal staff will be shutdown and unaccessible including those which has a booth or paid access. Non-regulated public lands are not affected by the government shutdown.
National parks are closed, including Yosemite. National Forests are still 'open' and so are state and county parks.
Services are closed and people with camping permits are being asked to leave. Hikers with backcountry permits are allowed to stay until the permit expires.
Another resource is to take a look at contingency plan for the National Park Service.
As of today, several states have committed funding to open federal lands (mostly national parks) temporarily in their states. Utah has opened all of its national parks and some monuments operated by the NPS.
Also, New York has paid to operate the Statue of Liberty park, Arizona has funded Grand Canyon National Park, and South Dakota is opening Mt. Rushmore.