BlueScreen and then memtest86 crashed, time to buy a new PC?

I think my computer has a hardware problem. It's a 5 years old Dell XPS 8100, the same one which had shutdown problems 3 months ago (by the way, those problems just stopped happening, magically I guess...).

3 days ago, I had a blue screen crash. When it happened, I was just watching some livestream video in fullscreen mode, on Google Chrome x64, using the built-in Flash plugin. The video was playing just fine, and the next second, with no warning at all, the screen went blue, it started saving the dumpfile, finished, and rebooted itself.

The error code in the Windows event log is 0x0000001e (0xffffffffc0000005, 0xfffff800030a331a, 0x0000000000000000, 0xffffffffffffffff). Quite helpful, right? Google took me to this forum thread so I downloaded BlueScreenView, which applied to my dumpfile, shows this KMODE_EXCEPTION_NOT_HANDLED in hal.dll:

Still not quite helfpul, really, the bug check string couldn't be more vague. Anyway, since it seems this might be a hardware problem, I downloaded the latest memtest86 version and created a bootable USB stick.

That was yesterday. Today I've booted the PC directly into the USB, and let it run the memtest86 with the default settings. Please note it's memtest86 v4.3.7; apparently my computer doesn't support the newer one.

The first time, it has taken 41 minutes to complete all the tests, and found 0 errors. I've read that it may take a few sweeps to find problems, so I've let it continue running for the second time. After around 1 hour more, it had completed 40-45% of the test (it seems to take more iterations the second time), and... the computer just rebooted itself again!

I don't know what to think. Am I having power-supply-electrical-whatever problems again? Was that a memory problem? Did the program just crash? It didn't show any message or anything, and the found errors count was still at 0.

What can I do? Is there any way to diagnose or further analyze any of these problems?

While it could be another power supply issue, it could very well be an over heating issue. The error with Windows was an unhandled exception, but since it is resetting while using Memtest, it rules out Windows altogether and there no need to go further with that for now. It's possible that your processor could be over heating and the system is automatically rebooting in an attempt to save it. Make sure that there's no dust on the heatsink of the CPU, or really, anywhere else in the computer at all, (take the machine outside before blowing it out).

Computers need to dissipate a lot of heat nowadays, so if there's dust built up around certain areas, it will block the heat from transferring into the air around it. This adds up quickly and will cause components to fail. Some of them, such as the CPU, are monitored by heat sensors and can trigger a reboot automatically. The power supply is no different either, and any dust inside of there might trigger the restart.

I say most of this in speculation, but it's backed up by some of the information provided: Both Windows and Memtest cause the reboots, so that rules out the software for the most part. Since dust acts like a blanket, one of the thermal gauges on the motherboard could be retaining it's own heat. If you open up the BIOS and leave it at the temperature monitoring page for a while, see if any of the temperatures go crazy. If they don't, then my guess would be that things are starting to point to the RAM. If you have multiple RAM sticks, then remove them one at a time, (ground yourself first) and let Memtest run again. If you have just a single stick, find another computer you can borrow one from and test that, (this is an older system, so I wouldn't buy RAM specifically for this system). If it's still having issues, then it could be something on the motherboard, in which case it probably is time to replace it.