I don't know if it's Oblivion specifically, but:

So for the past couple of weeks I've been playing Oblivion (with about 60 mods, of course). Except sometimes my PC shuts down in the middle of playing, and I have to wait a few minutes before I'm able to turn it on again.

I was wondering if it was overheating. I tried opening my case and use some dusting spray on the inside, but I still got shutdowns afterwards.

I installed MSI Afterburner to monitor my temperatures. My temperatures seem to only be getting as high as 50 C though, so I'm not sure if it's truly overheating or something else.

That said, my PC is about four years old so maybe it's just wearing down. And I've ordered parts for a new PC anyway.

Are you monitoring your GPU temps, or just your CPU temps?
50C wouldn't cause a shutdown.

I was monitoring both my CPU and GPU temps.

My specs are:

Gigabyte Radeon HD 7870 GPU with 2 GB RAM
Intel Core i5-4570 3.20 GHz CPU
8 GB RAM
Asus H81M-A motherboard
Seasonic G-550 Gold 550W PSU

Your situation seems to be classic example of PSU failure. It is always sad when such things happen.

Agreed, sounds like PSU failure to me too. One way to confirm it is to create the same (or higher) power load on the machine using some stress software. For example I use RealBench from ASUS which will fully load all your CPU cores and the GPU at the same time thus creating (close to) maximum power load on your machine and if you experience PSU issues it should similarly shut it down like Oblivion did.

Do note that if you do have PSU issues, putting high load on the machine risks permanently damaging components. Hopefully your PSU has protection circuits that will blow before anything too bad happens to the rest of the components but there's always the risk that won't happen. So I take no responsability if the above stress test ends up breaking your hardware.

Ah, too bad then.

Well as I said my computer is already four years old, and it was the first PC I ever built myself. And as I said I ordered parts for a new PC anyway, so I have a Ryzen + GTX 1070 build to look forward to. Maybe it'll even run Oblivion.

Unfortunately Oblivion (with mods) is a CPU bottlenecked game (and doesn't really use more than one core) so all those new cores and fancy GPU won't help that much :( What it does help is to have very high frequency per core (if you can overclock or if your CPU has some Turbo mode where it shuts down some cores in order to boost frequency of the remaining ones). But at least it shouldn't shutdown your computer heh.

The Ryzen and GTX was meant to run Witcher 3 at 60 FPS more than anything else. A new not-dying PSU is the bonus.

Though I guess I'll be able to run Oblivion in windowed mode without a hassle now?