Ground-fault circuit interruption (GFCI) receptacles, are not wired the same as regular duplex receptacles. In a standard duplex receptacle, both receptacles and all terminals are directly connected together (Unless modified). If one half of the receptacle is powered, then the other half is as well. With a GFCI receptacle there are LINE side terminals and LOAD side terminals, which are separated by an internal switching mechanism.

The wires feeding the circuit are connected to the LINE terminals, which supplies power to the device. If everything is wired correctly, there are not ground-faults, and the device is not tripped, then electricity is allowed to flow to the receptacles on the device and to the LOAD terminals. So if everything is functioning as it should, there should be be power at both the LINE and LOAD terminals. However, if the GFCI device is tripped, there will only be power at the LINE terminals.

Resetting the GFCI

If the GFCI has tripped, it can usually be reset simply by pressing the RESET button. If you press the RESET button and don't feel/hear a click and/or the button doesn't stay in, it means there is a problem and the internal mechanism is not allowing the GFCI to be reset. You can try pressing the TEST button, then pressing the RESET button again making sure you press the RESET button all the way in. If the device still will not reset, you'll have to try and determine the reason.

Why won't a GFCI device reset?

There are three reasons a GFCI device will not reset.

Wiring is wrong

If the GFCI device is not wired properly (LINE and LOAD reversed, hot and neutral reversed, etc.), the device will not allow a reset.

There is a ground-fault

Obviously, if there is a ground fault, the device will trip as soon as you try to reset it.

There is a problem with the device

If the device has gone bad, it will (should) not reset. Some devices will continue to hold, even if there is something wrong internally. However once they trip, they cannot be reset. Other devices will trip as soon as something internal dies, and will not reset. This is why monthly testing is suggested. If you press the TEST button, and then are unable to reset the device. You'll be made aware of a problem sooner, and can have it repaired (hopefully) before any damage is done.

Rewiring a new device

Before you begin, turn off the power at the fuse/breaker box and make sure it's off.

Locate the supply wire pair

There should be an ungrounded (hot) and grounded (neutral) conductor pair (likely as part of a cable assembly), that supplies power to the circuit. As it sounds like you've already located these, I won't go into detail as to how to locate them here (there are many other answers on the site that explain this procedure).

Terminate the supply wire pair

• Connect the bare/green grounding conductor to the green grounding screw on the receptacle (and to the box if required), and to any other bare/green grounding conductors.
• Connect the (white) grounded (neutral) conductor to the silver colored screw terminal labeled LINE on the device.
• Connect the (black) ungrounded (hot) conductor to the brass colored screw terminal labeled LINE on the device.

Terminate load side wires

If there are devices downstream that require GFCI protection, you'll have to connect the wires feeding those devices to the LOAD side terminals on the device.

• Connect the bare/green grounding conductor to the grounding conductors in the box.
• Connect the (white) grounded (neutral) conductor to the silver screw terminal labeled LOAD on the device.
• Connect the (black) ungrounded (hot) conductor to the brass screw terminal labeled LOAD on the device.

Set the device

Once all the wires are connected, install the device in the box using the mounting screws. Install the face plate, and turn the power back on.

1. Press the RESET button.
   • You should have power to both receptacles, and any downstream devices.
2. Press the TEST button.
   • You should no longer have power at the receptacles, or any downstream devices.
3. Press the RESET button again.
   
   • Power should be restored to the receptacles, and downstream devices.

The best solution here, would be to find a UL approved device that can satisfy your needs. Building your own device can be risky, and fraught with unexpected dangers. There are tons of UL approved power strips available on the market, it hard to imagine you wouldn't be able to find one that suits your needs.

If you must build your own solution, you'll want to be aware of at least a few of the dangers.

Exposed wiring

First and foremost, you'll want to make sure nobody can ever touch anything that could be energized. This means that all connections, splices, junction, etc. should be contained within an enclosure. You'll also want to make sure the enclosure is listed for the use, so that it will also contain any sparks, heat, etc. that might be produced.

Most areas now require tamper resistant receptacles in living spaces (living rooms, bedrooms, dining rooms, etc.), so you'll likely want to use TR receptacles for this project.

Overcurrent

While there are no hard-and-fast limits on the number of receptacles on a circuit, or the number of devices that can be fed by a receptacle. You'll want to be cautious not to overload your wiring. In industrial and commercial settings, 180 Volt-Amperes is the value that's used to calculate receptacle loads. If you use this as a guide, you should have no problems with overloading the circuit.

While the circuit breaker should protect the house wiring, you may want to protect the wiring in your cabinet from overload as well. You can do this with a fuse, or circuit breaker. You'll want to wire this in as the first device, possibly putting it in the same enclosure as the watt meter.

Surge protection

Since you'll be dealing with expensive electronics, you'll probably want some form of surge protection. There are receptacles available that offer surge protection built in, but they may not offer the level of protection you require, or that can be found in a manufactured surge strip.