Well first of all there is more than one kind of "safety." There are guns that don't have a switch on the side that says "go bang" and "don't go bang." Glocks, for example.

A Glock pistol doesn't have a manual safety. It has a drop safety, which is a little tab that prevents the firing pin from going forward that is moved out of the way by the pull of the trigger. That way there's no way you can drop, hit or shake the gun to make it go off without actually pulling the trigger. If you look at the trigger of a Glock, it looks like there's two; like there's a second trigger that sticks out of the first one. You see this on power tools too, it's a little lever that prevents the trigger from being pulled unless you first push that out of the way. That makes it a lot less likely to fire if you brush the trigger against something; you have to put your finger in the trigger guard, push the trigger safety down out of the way, and then pull the trigger.

Also, the way a Glock works, you can't load a round into the chamber without cocking the action, and you can't decock the action without pulling the trigger and firing the gun. (assuming no ammunition malfunctions here) When the gun is cocked, the trigger snaps forward, when the gun isn't cocked, the trigger stays back. It is common practice when carrying a Glock to carry it with a full magazine, an empty chamber and the action uncocked. With no cartridge in the chamber and the action uncocked, trying to pull the trigger won't do anything because it's already "back". You'd need to pull the slide back to cock the gun and chamber a round, then it'll go bang. If you've fired a couple rounds, and the chamber is loaded and the gun is cocked, the way you return it to the carry state is to remove the magazine, pull back the slide to eject the round in the chamber, point the gun in a safe direction and pull the trigger to dry fire the gun.

On the more primitive side, you have single-action revolvers. A single-action only revolver means the trigger ONLY does the job of releasing the hammer so it can fire the cartridge. If the hammer is forward, it has to be pulled back with the user's thumb or other hand to cock the action and rotate the cylinder to the next chamber. There's no need for a lever on the side of the gun because you already need to fiddle with a lever on the back of the gun. If the gun has been recently fired, the hammer will be resting on a spent cartridge. I have heard some say it is good practice to carry such a gun with the hammer resting on an empty chamber, which is basically the same idea as the Glock above; you're loaded with one fewer round than the absolute maximum but carrying in a way where there's no bullet aligned with the barrel and ready to go.

Have you ever heard of a revolver? Most of those do not, and are one of the oldest designs still in use.