I don't have 60D, but some cameras also have "trap focus" feature which will not let the shutter fire until the focus is achieved (this is actually quite handy for some types of photography, such as manual focus macro). Most of the cameras always do this automatically in One Shot, the green LED in the viewfinder blinks to show that focus wasn't achieved. I think some models have an override setting, you can set so the camera will go ahead and take the shot anyway. Or use AI Servo. You can test this by trying to focus on something that's closer than your lens' minimum focus distance. Just set the camera to One Shot first. It then should work in AF or even if the lens' AF switch is turned off and you are manual focusing.
I don't know if the 60D uses the same type of shutter release button as the 50D and 40D. But, yes, as noted in a couple previous responses, those are known to get "gunked up" from finger oils and such over time, can be "flood cleaned" with isopropyl (rubbing) alcohol... carefully... without disassembling the camera. Disassembly and a proper cleaning would be better, if possible. Usually the switch just needs cleaning, not replacement (though Canon service dept. would probably just replace it in any case).
Third possibility is some sort of lock that prevents incorrect exposure, though I'm skeptical about this. Usually it's a "Safety Shift", which when enabled will override your ISO or other settings if too low or high for a shot, but will go ahead and take the shot. I.e., it doesn't prevent taking a shot, even when Safety Shift is off... but maybe it's different on 60D.
You didn't mention what exposure mode you had set. Some of the highly automated settings (i.e., the Mode Dial icons such as running man, mountain, etc.) go way beyond simple exposure settings and lock certain things on the camera.... maybe one of those is causing an exposure or focus lock.
Also check that the battery door and the memory card cover are securely latched in place. There are micro switches on those on most Canon cameras, to prevent the camera from working if they are ajar. Sometimes dirt or whatever can get in and cause those switches to jam, too. Usually the entire camera is dead, though, if this is the fault.
There's also a micro switch under one of the springs in the flash mount, that sometimes gets stuck so the camera thinks there is a flash mounted. Usually the symptom is that the built-in flash won't pop up and an error message is displayed, but if the camera is waiting for a flash to recharge, and there is none in the hot shoe, it might prevent the camera from firing. If this micro switch gets stuck, it's often posssible to coax back with a wooden toothpick or the point of an Xacto knife. Also might need to slightly expand the spring to prevent it from stopping the micro switch from fully opening. This is just a possibility.
Turning the camera off and back on clearing the problem is a bit of a clue... Though I'm not sure of what. It's almost as if the camera is in some other mode that doesn't allow images to be taken, such as accidentally having set mirror lockup (in which you have to press the shutter release once to lift the mirror, then a second time to take the shot).
And, if removing the rechargeable battery(ies) and then reinserting fixes the problem temporarily, it might be a low voltage issue that can be cured buy carefully cleaning the battery contacts (just wipe with a clean rag dampened with a couple drops of isopropyl/rubbing alcohol). If using a battery grip, also clean the grip to camera contacts, as well as the battery to grip contacts, and make sure the grip is snugly fitted to the camera.
Fnally, use the same clean rag/alcohol technique to clean the lens to camera contacts... It might be that the two aren't communicating with each other properly.
