Hmm, where to start?
In my experience, some people are better able to withstand the pressures of stress and exhaustion than others. To a certain extent, tolerance to and performance while under stress and exhaustion can be trained into a person (combat military and doctors, for example). But unless you have actually been so trained, I'd advise against pushing it, especially with something as potentially catastrophic as having a firearm on-hand. Or driving a vehicle. You just may have to "call in sick" and take a "power-sleep". And it wouldn't be a lie -- a body requires rest, it isn't optional. Look after yourself.
As far as being a hazard to yourself or others by having guns on-hand: As long as you are actually going to get some sleep pronto-like, I wouldn't worry about it too much. You will experience some level of decreased situational awareness while you are in deep-exhaustion sleep, even more so than "normal" sleep. With that in mind you might want to arrange for some alternative security precautions for the time being, if your situation really warrants it. But get some sleep. Nobody -- not even doctors who train for this sort of thing -- can keep burning the candle at both ends indefinately.