Unable to boot into any mode...storage devices not recognized

Most likely a power supply failure, that's a non-servicable part except by the manufacturer, so it's one you replace. [can hold a charge and possibly KILL you if opened up]

Any PC repair shop near you can verify that the PSU is croaked in a minute, with a PSU tester which you can buy for $20ish.

Rosewill RTK-PST Digital LCD Power Supply Tester - Newegg.com

PC is acting like it doesn't have the juice to run the parts, just so much more likely PSU dead, eliminate that before moving on, hopefully it didn't damage anything else.

The main power supply unit in the case that powers EVERYTHING.

It's suspect since you said the system basically croaks shortly after looking like it MIGHT work and not recognizing things.

Otherwise, 2nd suspect is your motherboard and which one would you rather verify is bad, first?

Try another power supply to your wall outlet cable and make sure your surge protector or power strip is not dead. [less likely]

Even multimeter test outlets for proper voltage range, there is so much possibly going on besides just not having it see a drive.

You said you couldn't even get into the BIOS, but it sorta of started on another reboot and started to take a rescue disc but then stopped.

You care to start your description over?

Where are you seeing not recognizing drives?

Power supplies can work and overheat and then stop working... turn it off a bit do that again, same result if that is the case.

The performance of PSUs degrades over time, they can't handle as much as they might be listed for, not all components are on full draw on the PSU at boot either, like video cards.

3am here and if I misread something, I'm sorry, I'm fatigued, but that's what I gathered from your posts.

The usual failure points on PCs: PSU, RAM, motherboard, Hard drives, and video card, only the PSU and motherboard fits your description of symptoms

My last failure was my motherboard, symptoms: couldn't do/see ANYTHING.

Replace your CMOS battery (CR-2032) on your motherboard is something I would do first, without that, BIOS is in a default state, let's see if you can get into the bios then after that.

Looks like HP is taking you to their own recovery console, likely on a hidden hard drive partition.

I'm hoping it's the $3 CMOS battery now for your sake.