Unplug the system, remove the battery and hold the power button for 30 sec. Remove and reinstall the memory modules.
Power up on AC only - no battery. If the system will power up, but won't with the battery installed, the battery is bad. If it still won't power up, the mainboard is bad - it's up to you whether to have that replaced (it'll be a minimum of a couple of hundred dollars), but given the system's age and low-end nature, a replacement system is probably a better idea.