Just trying to imagine how you might exclude a malignant government from your transmissions in a particular type of national emergency.
You are correct. Generally repeaters operate on the 2 meter VHF band although there are also numerous 70 cm and 6 meter repeaters out there. A repeater receives on one frequency and simultaneously transmits on another. VHF radios automatically make the shift adjustments on send and receive.
Repeaters are placed on high points, often sharing towers with commercial and public safety equipment. For the most part the repeaters are available for general use. However, most are also designated as emergency resources and if there is an emergency general traffic is disallowed.
That said, most hams use other methods (HF bands, relays, etc.) so that they can work effectively without having to rely on the "official" infrastructure.