My first thoughts are overheating power supply, ssd failure, which is not impossible by the way, or a failing motherboard. Sorry for the grim forcast, but this stuff happens to folks all the time.

Hi cx2
I have this on my brother's i3.
Either the board or the power supply itself could be on the blink.
or the cmos battery.
What I'd do is clean the dust out the system.
Clean all the components, and clean all the dust with compressed air either with a compresser or something or whatever.
I'd also clean the heatsinc with the compressed air and regrease the sync.
I'd also check all fans are spinning as I have known to have fans stuck.
I'd reseat the cmos battery or even replace it.
I'd check the bios smart info I assume you can see to build your own system.
I'd also run pcpitstop, from what you are telling me it sounds like the board or cpu may be out.
I was able to however after removing and cleaning all fans and such clean the  cpu and other things and blow the dust out the back.
I then cleaned the sync and removed and regreased the sync.
The issue is still there but its greatly reduced.
The other thing I suggest is getting something like motherboard moniter and monitoring the temperatures, maybe there is a fan failier in the system and things are going way hot.
As ffor activations is this spreading beyond jaws.
If cards are doing strange things or failing to start it does sound like a failing component, the main board or processer.
I have had graphics cards and tv cards fail on occation.
At any rate regrease the heatsync check and if needed replace all fans and regerase all syncs and clean your dust and it may work.
I did this and the system has not crashed though that doesn't mean there isn't still a problem with my power supply.
I'd also suggest to not hibernate or sleep the system to minimise stress on standbuy and to remove unnecessary programs from loading.
How old is your system?
It could be time to replace it.
Systems to get dusty from being used naturally.
The systems here including my laptop and the 2 desktop workstations are used daily sometimes they crunch numbers all night.
I had an old amd system run for 6 years doing various things before a card cooked itself and that was with 24 hour use.

I am having a similar issue with my windows 8 laptop that is about 2 years old.
A lot of the time, when I shut my computer completely down and turn it back on, or when it is restarting, it will just shut off in the middle of restarting. When I turn it back on it will be on for 10 or 15 seconds and then just turn off again. I have to turn it on like 5 or 6 times before it actually boots up properly. Because of this I put my computer on standby when i'm not using it for short periods of time.
I've also noticed that after a very short time of using it, it feels extremely hot, and the fan is also constantly on and extremely loud. Do you guys think it's a fan issue or something more serious? I can't figure out why my computer is constantly turning off when I try to turn it on, so I always leave it on standby.

Mine was on for more than 30 seconds before I realised the fan was a problem, turns out the blades were stuck on the fan's power wire. Unfortunately since then I've found my processor temperature going up to the 70 degrees region just with light web browsing for an hour or so. I'm probably going to replace the CPU cooler but I'm concerned if it might be something wrong with the processor since its overheating previously making its temperature rise, I'm loathed to spend the money on a new cooler if it won't help but I don't see any other way to find out.