He's had it since launch, so the warranty is long since expired.
(edit) Check HDD first. It's possible you have a dead drive if you're lucky. If you don't have an extra laying around like most normal people, try pulling it out, cleaning off the contacts and the HDD port and plug it back in. Won't do anything if the drive is dead, but in some rare cases, it could just be a faulty HDD connection.
Most likely the problem is with the motherboard, although the PSU (shouldn't power up at all) and the BD drive can also fail (a console with a broken BD drive will still start up; just won't read discs).
Ask yourself how important PS2 BC is. If the answer is not very, consider a Slim as they use less power, run cooler and are quieter (slower fan speeds). Generally those three things mean better reliability.
As far as getting a refurb from SCE or a repair if you really want that PS2 BC, I'm not sure if they still have replacements, assuming they can't repair your console. Give SCE customer service a call and find out.
If you don't want to go the $150+ handling charges for the repair/refurb, you have the option of attempting to repair the console yourself by reflowing the GPU and CPU. Tutorials are online. All you need is some good thermal interface material, some non-corrosive solder flux and a heat gun. If you're not handy with PC components, you should be able to find someone local who can do the job for you.
The caveat is that once you crack that seal, even if it's just to clean the dust out of the case, SCE is no longer obligated to accept your console for repairs; you're on your own at that point. Caveat number two is that the reflow fix should not be considered a permanent fix unless you do a really clean job, use high quality TIM, use soldering flux before reflowing, mixed with a little bit of good fortune.
I'd recommend using a shop vac with hose attachment to vacuum out the vents before cracking the case open. I'm going to second the recommendation against using compressed air. Yes, it will blast out dust balls, but it can also blast dust into components as well.
If your console has been running louder than usual the last few weeks or longer, depending on how much you use it, you may have already damaged your motherboard if the cooling system has been too clogged up to function properly and removing the debris won't magically bring it back to life.
