The basics: A VOR (Very high frequency omni-directional range) station just gives you the bearing to the VOR. It's simple. It's a large ring of antennas with another antenna in the middle. It sends out a big omnidirectional pulse, and then sweeps around the circle like a lighthouse. The time difference between the omnidirectional pulse and the directional pulse tells you your bearing to the VOR station. The aircraft just receives; it doesn't send anything. Range is maybe 200 miles.
DME (Distance Measuring Equipment) came later. It's a request-response system. Time between aircraft request and DME station response gives you distance to the DME station. Most VORs also have a DME system installed, so you can get range and bearing.
VOR bearings aren't very accurate. Error is up to ±4°. So position from VOR and DME isn't very good far from a VOR. VORs are thus installed at major airports, so positional info gets better as you approach the airport, and pilots can find the airport reliably.
SJC (San Jose International Airport) has a VOR northwest of the airport. It's a huge antenna array in a big open field, and can be seen from 101 north of the airport. It needs all that open land to work well. Obstacles would distort the directional beam and make the error worse.
The FAA has shut down over a hundred VOR stations as redundant.[1] The original plan was to shut down even more, but there was much pushback. In addition to airport VOR stations, there were chains of "enroute" VOR stations, so that aircraft could fly along established airways from VOR to VOR. Some of those have been shut down.
The FAA now uses the term "minimum operational network" for what's available with GPS down.[2]
GPS jamming is very real. Here's a real-time map of known GPS jamming and spoofing.[3] Current jamming is mostly near Ukraine and Lebanon, plus the Black Sea. War zones. Discussion at Ops.group, which is a site for people involved in international aviation operations.[4]
Edit: sources provided are helpful, thank you lsh123 and Animats!