First thing to do is to unplug the system from the wall, wait about 30 seconds, then plug it back in to see if that will allow it to turn on again. If the unit shutdown due to a sudden loss of AC power, you'll have to unplug the system from the wall to reset it before you can turn it back on again.
If that didn't work, unplug the system from the wall again, find the CMOS battery on the motherboard, and remove it (you might need an owner's manual for instructions on how to do this); leave it out while you strip the system down. Doing this will clear the CMOS settings, which might have been corrupted, and reset them to the system defaults.
Now, unplug just about everything from the motherboard; disconnect ALL of the drives, including the floppy drive. Remove BOTH the power (MOLEX) plugs AND the data / ribbon cables. I did some research; according to the eMachines website, your system uses onboard sound, NIC, and video, though I think the modem is removable, so remove it if possible; disconnect the speakers; remove the mouse and any printer / scanner / USB peripheral you might have attached. Disconnect any network / modem / USB / FireWire cables from the front and back of the system. Disconnect any USB hubs or thumbdrives, external burners, ZIP drives, parallel port devices, external fax/modems... you get the idea.
If you have the optional flash-media reader, disconnect it. Disconnect the PSU ATX connector from the motherboard, and check it to be sure that it isn't burned or damaged. Check the ATX socket on the motherboard as well; if it looks OK, plug it back into the motherboard. Be SURE it is FULLY seated in the socket on the motherboard; push down on it evenly until you are positive that it is making contact all along the length of the connector. Remove the RAM, then check the pins on the edge connector for tarnish or corrosion. You can clean the little gold pins on the RAM with 99% isopropyl alcohol (isopropanol), also known as "rubbing alcohol", or methyl alcohol (methanol), also known as "wood alcohol", but use the stuff wisely, or avoid it completely.
At this point, you should have only the keyboard, the CPU and CPU fan (check to be sure the HSF fan wires are properly attached to the motherboard), the PSU ATX connector, one RAM DIMM, and the monitor cable attached to the system. Every other piece of hardware should be disconnected (IDE drives / USB / FireWire devices) or physically removed (PCI/ISA cards). Again, be sure to disconnect both the power and data cables from the HDD / optical / floppy drives. You want to strip the system to the bare minimum required to make it begin the POST process; you NEED the CPU, one stick of RAM, the videochip, and a known-good PS/2 keyboard couldn't hurt, if you have an extra one available. Double-check to make sure the keyboard is plugged into the proper PS/2 keyboard port; it will NOT work if you plug it into the mouse port.
Next, install the CMOS battery, but be SURE you put it in properly; you don't want to cause more problems by putting the CMOS battery in upside-down. Check everything one more time; is the video cable from the monitor attached to the back of the system? Is the PSU ATX connector fully seated on the motherboard? Is the CPU fan plugged into the motherboard? Check CAREFULLY; if you don't have the CPU fan plugged into the motherboard, you could destroy the CPU. Is the RAM properly seated in the DIMM slot? Check everything carefully, then check it again to be sure...
When you're POSITIVE that you have everything connected properly, plug the system into the wall, and try to turn the computer on. Pay close attention to what happens; if you see smoke, or smell anything burning, pull the plug out of the wall and check the system for damage. Listen to the system; is it making any unusual noises? Is it beeping? If it is, write down the sequence of the beeps (two short beeps, four long beeps, or whatever it might be). If everything is working as expected, you should see the eMachines logo on the screen, or see the POST process begin. If the system begins to POST but then stops, check for any error messages onscreen; you EXPECT to see error messages, because you've disconnected the HDD, floppy, and optical drives, so the system will NOT be able to finish booting.
At this point in time, you should know whether the system is now bootable (so you can shut it down, unplug it from the wall, and put it back together), or whether you still have a problem. Either way, let us know what you've learned...
