Welcome 'esjbh2'
It could be as simple as a failed momentary switch on the front panel of the case.
The Delta 297-watt power supply is a proprietary 18-pin main motherboard configuration (vs. standard 20+4 or 24-pin ATX, Micro-ATX, Flex-ATX configurations). Before heading down the road of replacement (most likely that means getting one from Sony or an on-line supplier) let me ask you a few questions, starting with the obvious process of elimination questions (sometimes it's the easy ones that trip us up):.
1). When you have the lid of the case off, and the power cable connected to the Power Supply Unit (PSU) and plugged into a powered outlet, do you see a lighted, green LED on the motherboard near the button-cell battery? If that LED is lighted, you have continuity from the wall outlet through the PSU to the motherboard.
2). Have you pulled the power cable from the wall outlet and plugged another electrical device into that outlet to verify you do not have a failed wall receptacle?
3). Do you have your VGX-XL3 plugged into a surge-suppressor or other power strip; and if so have you checked to see if the circuit breaker has tripped on the surge suppressor or that the plug-strip has not failed.
4) Had you made any alterations, upgrades, modifications just prior to the power issue?
Cheers!
Robert.
1) No lighted LED. I see the LED you're referring to, but it's not lighting up.
2 & 3) Yes, I've confirmed power to the PSU using various reality checks.
4) No recent alterations, upgrades, or mods.
I looked around inside the case hoping to see a fuse to replace or similar, but saw nothing. So I assume there's nothing simple like a fuse or circuit breaker on this unit?
Additional information which may or may not be related???....For the last month or so, I had been getting a checksum error on restarts. It would keep auto restarts from being successful, but it would always start when I hit the F key to continue at the prompt, so it was seemingly just an annoyance. Some online research suggested either replacing the button-cell battery or flashing the BIOS could address issue. I was going to take action on this but found that the unit was powered off & DOA and here I am.
-Jack
"Hi Jack" ( not a smart greeting while boarding a plane ).
Well, it's good that you have eliminated the easily overlooked stuff.
Because you have no standby power to the board (a dark LED) that suggests no standby power from the Power Supply (or a dead LED; possible but that's pretty rare). I would suspect the power supply at this point.
I have not opened the Delta Power supply in my VGX-XL2, but I would be quite surprised if there is NOT a replaceable fuse inside it. That's probably a good next step for you, to pull the power supply.
Just be careful to make note of what everything connects to, as you disconnect each of the power supply leads ( I always cut strips of paper and write my notes on those strips, and then tape them in rings around the leads to remind me what they attach to when I reassemble a system.
Once removed from the VGX-XL3 case you can crack open the power supply enclosure and scout for a removable fuse. It is probably a ceramic body fuse, but may be a glass body fuse. If it's a ceramic body you won't be able to see if it's blown. If you have a tester you can check the continuity. But gosh, they're cheap, so you can just assume it's blown and replace it. Just go to Radio Shack and get a replacement. Radio Shack is pretty well supplied in the common fuse sizes and amperage.
If you do find a replaceable fuse inside, and you replace it so you know it's good, it might also be a good idea to then take the power supply to a local computer repair shop and have it tested before you go to all the trouble to re-install it. If it turns out that the power supply is "toast", let us know and maybe one of our fellows will have a lead for you for getting a replacement. They're not terribly expensive, but like all things Sony (and all things proprietary in design), they cost more than standard units.
As for the BIOS, ....don't mess with it now, Jack. We don't presently have a backup copy of the original VGX-XL3 BIOS for you to re-flash it (we may shortly have that original as well as a BIOS update, but not quite yet).
If you bugger the BIOS right now, you're dead. You'll just turn your beautiful VGX-XL3 into a 'wheel chock' for the camping trailer! 
But ...it won't hurt to "reset" the BIOS to the OEM factory defaults by clearing the CMOS, and replacing the button-cell battery (I believe your system uses a standard 'CR2032' battery, but please verify that before you buy one.
The easiest way to clear the CMOS on a non-bootable system is to short the "CLEAR CMOS" jumper on your motherboard. On the "P5BW-MB" motherboard for the VGX-XL3 I believe the CMOS jumper is marked "CLRTC.".
CAUTION: This needs to be done when the system is not powered; disconnect the power cord just to be safe. Open your computer and look for the little 3-pin jumper near the button-cell battery. Only pin number 1 will be visible; the other two pins (2&3) are presently covered by the jumper. You clear the CMOS, (and reset the BIOS to the factory defaults) by moving the jumper from its pin 2&3 position to the pin 1& 2 position, and then move it back to the pins 2&3 position before reconnecting the power cord.
If you disconnect the power cord, and pull the battery for a period of time, that will also clear the CMOS without needing to short the jumper. Either will work.
Good luck, Jack. And keep us updated on your progress.
Cheers!
Robert.
Repair Parts :
(And I wonder what the 'back story' is accounting for the bloody fingerprints on the Delta power supply enclosure?) 
I suppose this confirms what I have been reading, that Windows 8 is proving to be a "big hit" with users after they have transitioned (which surprises the heck out of me, after playing with it for just a few, frustrating hours). I do want to transition, however, because time marches forward, and it's either keep current with Windows, switch to Apple (or say goodbye to the Robber Barons and go with 'Ubuntu' and 'XBMC').
