If you know the receptacles that you have plugged the computer into, indeed have power, I would suggest you diagnose the Power On switch, and the Power Supply.
See of the problem is the Power On switch, or the Power Supply.
A jumper wire is used on the main ATX power cable.
1) If the Power Supply comes on you have a bad Power On switch.
2) If the Power Supply does Not come on, you have a bad Power Supply.
Your problem is posted in the eMachines Bestec ATX-300-12E power supply section.
Going with that we are discussing a Bestec ATX 300 watt power supply, the main ATX power cable is a 20-pin ATX main power cable.
Typical pinout of a 20-pin ATX main power cable,http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain20
Note the Green wire in the BACK of the connector, in the photo to the far right.
This is the Soft Power On wire, or also abbreviated as the PS_ON wire.
Note the Black wires. ANY Black wire is a Ground wire.
With the 20-pin ATX power cable plugged into the motherboard, as shown in the photo, a jumper wire is touched to the Green wire, and to ANY Black wire.
The computer (Power Supply) is also plugged into power.
Generally a paper clip is used for the jumper wire.
The paper clip is straightened out, then bent into a U shape.
Black plastic electrical tape is then wound around the middle of the U shape.
This taped area is for your fingers, and thumb to hold onto.
The 'legs' of the paper clip, are then inserted down into the socket holes, in the back of the 20-pin ATX main power cable's connector.
Down into a socket hole with the Green wire, and also into a socket hole with ANY Black wire.
The 'leg' of the paper clip goes down into the socket hole, RIGHT NEXT to the existing wire.
At the end of every wire in the 20-pin ATX main power cable, is a metal terminal end.
The metal terminal end is pretty far down in the socket.
You must ensure that the metal end of the paper clip, touches the metal terminal.
The voltage is 5 Volts DC.
Two D cell flashlight batteries produce 3 Volts DC.
There may be a small spark as you make the connection.
Warning you in advance.
You can see the voltage is VERY low, however.
I would like you to perform this test, then post back in a Comment as to your findings.