Some old designs (think antique revolvers and some antique semiautos) could go off if they dropped and fell on the hammer a certain way, or some older semiautos fell pointing straight down so that inertia could move the firing pin hard enough to discharge a shot into the ground on impact. In the 19th century, revolvers were often carried with an empty chamber under the hammer to render them drop-safe, but that is not necessary with most modern designs. Modern handguns incorporate passive safeties to prevent accidental discharge, and most could be thrown off a building onto pavement and not go off, though some mid-century designs (like the Walther PPK and the Phoenix Arms Raven) require the manual safety to be on.
Ammo can go off due to heat if the temperature of the powder exceeded its ignition temperature, but that would be around 300 degrees, give or take 10%. You can achieve that in an oven, or in a fire, but not by leaving it in a car with the windows up.
Nearly 100% of the "the gun just went off" stories you'll hear involve accidentally pulling the trigger, usually either as the result of carelessly putting a finger on the trigger when the shooter did not intend to fire the gun, trying to catch a falling gun (don't, it's drop-safe but a snatch can pull the trigger), or accidentally pulling the trigger while holstering. It's a fundamental rule of gun safety that your finger goes on the trigger only when you are about to shoot, and one of the easiest ways to identify a total n00b is to see whether they casually rest their finger on the trigger.
Some older bolt-action rifles designed in the 1940s through the 1970s aren't drop-safe because they weren't intended to be carried with the chamber loaded, but most modern designs incorporate better safety systems and are intended to be drop-safe with the safety engaged, and some incorporate passive safeties as well. Also, some older rifles and shotguns could be fired by closing the action, and the Remington 700 had a problem for a while in which a neglected and dirty rifle could fire when the safety was disengaged, but handguns designed to be carried loaded tended to have a lot more attention paid to those kinds of failure modes.
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