The smell is the Power Supply.
Typical example of a Power Supply not installed in a desktop computer,http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=4435072&CatId=106
This is not an example of a direct replacement Power Supply for your computer. I would need to know the Model Number of your eMachines to give exact information ]
Bad Power Supply. Weak voltage power rail. (Or rails. There are three main voltage power rails for a Power Supply used in an eMachines desktop computer)
Enough power to light LED lights, and apparently spin fans, but Not enough to turn the Processor on.
Watts = Voltage times Amperage.
One measurement used for measuring the maximum available power output, of a desktop Power Supply, (SMPS. Switched-Mode Power Supply), is to state the Maximum Wattage available.
Typical maximum Wattage available for a typical eMachines Power Supply, usually averages from 150 to 300Watts.
1) ALL of the LED lights on at one time, will use less than 1 Watt of power.
2) EACH fan uses 2 to 3 Watts of power.
3) A typical Processor can use 51 to 125 Watts of power.
Depends on what Processor it is.
(Note* Intel Pentium II processors use less than 51 Watts)
Solution is to replace the Power Supply.
This isn't as hard as it 'sounds', if you have never performed this procedure before.
I know you can do it.
If you need guidance in replacing the Power Supply in a step by step for your eMachines, please post in a Comment. (Believe upper right of your page)
Should you also desire recommendations to compare for a Power Supply, I would be happy to provide them.