There is no power switch on the enhance atx 1123b that I have . What actually powers it up?

Basic power supplies should be 400 to 1000 Watts for current systems. A 230W supply is Underpowered and may not support the hardware devices all that well. In addition, the Video drains power, so a Video card with 2G Ram will help speed up the process. Then an additional Power RAM (Memory) with speed available to your system; helps increase activity. Cooling Fans help reduce HEAT that also slow down operations. Clean the inside of the computer as dirt., dust and dust balls collect and keep the electronics hot.

Black=0V(GND)
Yellow=12V
Red=5V
Orange=3.3V
Purple=5Vsb(everytime on)
Gray is PWRGOOD or PWROK(pwr=power) must be at 5v
White=-5V(optional)
Blue=-12V
Green is PWRon must be connected to a black wire to turn on the PSU

The only thing to do at this point is to replace the Power Supply with a KNOWN to be good, compatible unit, and see if the rest of the computer hardware components are okay.

Going by your information stated that the Power Supply is an Enhance ENP-2224B SMPS, (Switched-Mode Power Supply), it is a special shape.

If it wasn't for the Slim Micro-ATX computer case you have, you could use an easy to obtain ATX form factor Power Supply.

From a quick search though it looks like a replacement unit is affordable,

http://www.censuspc.com/Enhance-ENP-2224B-2G-240W-SATA-TFX12V-P4-Power-Supply-for-Slim-microATX-Desktop-Case-Serial-ATA-Ready-bulk-pack-SKU-ENP-2224B-2G-pr-3033.html

There is simply no way to tell if the motherboard, and/or Processor, and/or Ram Memory, and/or Harddrive, etc., is okay without having power.
Unless you transfer the components on the motherboard to another working motherboard.

[NOTE*
If your Power Supply has a rocker switch on the back that lets you use 120 volts, or 220 volts, make SURE it is set to the right voltage.

USA?
120 Volts
Europe?
220 Volts
Australia?
220 Volts
Japan?
100 Volts ]

For additional questions please post in a Comment.
Regards,
joecoolvette

eMachines uses low quality hardware components inside their computers. This is the way they can sell them at budget prices.

One main hardware component is the low quality power supply they use.
A generic power supply made by Bestec. (A generic power supply manufacturer)

(PSU. Power Supply Unit. The type of PSU used is a SMPS.
Switched-Mode Power Supply,

http://en.wikipedia.org/wiki/Switched-mode_power_supply )

Low quality;
A) Electrolytic Capacitors (Radial Aluminum Electrolytic Capacitor)
B) Rectifier Bridge
C) MOSFET/s
D) Wiring is too small in gauge
to name a few of the components.

However IMHO it is also best to check out the Power On switch.
There is a simple test to see whether the problem is a faulty Power On switch, or a faulty Power Supply.

The test involves bypassing the Power On switch. The Power On switch is not involved in this test.
A jumper wire is used on the Soft Power On wire, [ PS-ON ], and a Ground wire in the ATX main power cable connector.

If you bypass, and the Power Supply comes on you have a bad Power On switch.
If the Power Supply does Not come on you have a bad Power Supply.
If you would like to know more in detail on how to perform this simple test, simply post so in a Comment.

The Power Supply used is just a common ATX version.
ATX refers to the Form Factor of the power supply.
Regular size ATX form factor is approximately 6 inches Wide, 5-1/2 inches Long, and 3-1/2 inches Tall. (Height)

There are also differences in the pinout of the ATX main power cable.
Yours may have 20-pins, or 24-pins.
Power Supply's sold now take this into consideration.
Their ATX main power cable can be used as a 20-pin, or a 24-pin version.

[ General example of a 20-pin ATX main power cable, and it's connector,

http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain20

Example of a 24-pin ATX main power cable, and it's connector,

http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain24

Your motherboard may also require an additional power cable, commonly referred to as a P4 power cable, or P4 ATX 12V power cable, or just ATX 12V.
The actual name is 4-pin ATX +12 Volt power cable,

http://www.playtool.com/pages/psuconnectors/connectors.html#atx12v4

If you would like help in finding an inexpensive reliable Power Supply, please post in a Comment.
I will need to know what Model Number of eMachine you have. It's on the back next to the Windows product key, or up on the side of the computer tower.

If you need anything explained that I have stated here, please let me know in a Comment.
Regards,
joecoolvette

This is just a normal ATX power supply.
ANY ATX power supply rated at 350W (Watts) or more will work fine.

If your ATX switch mode power supply is faulty then I suggest you replace it with a new one that is 350 watt or higher wattage.
Fixing faulty ATX power supplies is not a DIY project, there are dangerous voltages in these power supplies and can cause death or serious injuries.

The more amperage you have on the 12 volt rails the more power you can draw from the power supply. For an xbox 360 I don't see any reason why you would use a non-standard power supply unit there really is no gain using a computer PSU on an xbox 360 unless you broke your factory PSU and then you could rig a computer PSU to it but your better off not trying unless your really good at electronics.

Please change or replace your PS system.

Yes, you can safely put in equal or higher. Just make sure it is an ATX and that it is P4 compatible. Most new ones all meet to those specs. Higher is actually better. I would go with a 400w or higher.

The old AT power supply that these old computers uses has two six pin socket connectors and these are different to the ATX connector.
You cannot buy these power supply now, most likely you might find one/computer in a junk shop.
If you have electronic knowledge you might be able to cut off the connectors from the old power supply and connect it to the ATX power supply and modify the ATX power supply so you can switch on the power to the old computer. The old AT power supply uses a mains power switch whereas the ATX uses low voltage to turn on the power supply.
Good luck & have fun.