if you are trying to turn on the laptop with only battery power it may be a bad battery and you can turn it on by using your charger. Even if you don't have a hard drive it should still boot to your BIOS.
Usually when it's something like that I'd first check the battery, then the ac adapter, then it gets expensive and annoying. After that it's probably the power supply or the motherboard, at least in my experience, and since you're out of warranty and the model is really outdated, if it's not the ac adapter or the battery, it might be time to look at something new...
First remove battery and try to boot with adapter only
Check if the adapter LED is lid up.
Remove simple parts like HDD and ODD ( if its present )
Check if the memory is installed correctly
Remove ( if there is more then one memory bank ) 1 of the memory banks and try to boot. Also if still no effect try to remove the other memory bank and you also could try another memory slot.
Thanks. Got some spare time tinkering with an old laptop. I've already done all that's been suggested and I suspect a bad mobo. I tried powering up the motherboard outside the casing just to know which chip is creating the heat I felt when I applied power. It is the chip named ENE KB3920QF. It generates blistering heat a few seconds of applying power. So I guess the mobo's fried and repair is not a good advice anymore. Thanks again guys!