I'd check the power supply first (with a Volt meter?). When that's fine replace the motherboard. When in doubt, replace the power supply first, that's cheaper.
But it might be a very good idea - after 7 years - to buy another PC or laptop. You might find a good deal now or Monday. You'll be amazed how fast it is, especially if it has an SSD inside for the OS.
This problem just happened recently. When my network adapter driver stopped working, I restarted the computer. It didn't turn on. I didn't hear POST, or beeps. My computer didn't boot up. I tried again, the power indicator turned on, then off, then on again. I tried disconnecting everything:
the USBs, Audio, VGA, SATA Data and Power(for CD drive and HDD), and the power suply from the computer, then I reconnected it again. Nothing. I also reinstalled my CMOS battery(I usually boot without it), my RAM, I even tried using a recovery disk. It just got stuck inside the CD drive.
My specs are:
Motherboard- Gateway SX2800
Computer Model- Acer Aspire AX3810
Computer Age- Old, about 7-8 years