DO NOT TRY THIS AT HOME!
If you want to try this at home:
you can just buy a kit that's so much safer!
If you want a passable action sequence that will put out a tiny bit of EMP but you can make in under 60 seconds - that's at the end.
If you actually want to disarm a bomb be aware depending on design it may blow the detonator as it fries the circuitry. I'm assuming your character has inspected the bomb and realised that an EMP is the right choice.
You can build an EMP gun from parts in a server power supply and wall socket cable.
This inside view of a power supply:
Those two blue capacitors tend to be quite high capacity and low inductance. Each server will have 2 power supplies minimum, and each power supply will have several large capacitors. Read the voltage on the side. Collect at least 250V worth, ideally 500V worth.
Grab some insulated, single core wire. Speaker cable is ideal, an extension cord will do. Cut both plugs off, strip the wires, and join all the wires together (positive / negative). Twist into a ring about 20cm in diamter, and then use duct tape or something to hold the ring.
Youre coil will look like this:
Note both conductors are joined to make one big effective cable that loops.
Now build your capacitor chain. Read the voltage on the side, and wire them in series up to over 250 volts. (You may need to find a soldering iron in the server room somehow). If you have more than 250V, build two sets, and solder the strings of capaictors together in parallel.
Wrap with duct tape to stop them wiggling and breaking the soldering.
To build a charger, you need a bridge rectifier (or diodes) and more capacitors. The circuitry is very common; it can look like this:
Or you can get it prepackaged, it comes in dozens of forms:
Any of these will do so long as they're rated over 240V.
Grab a power cord and strip the connector off, leaving the wall plug and some cable.
Charging and use:
To charge - Connect bridge rectifier DC side to your capacitors (matching positive for positive). Then connect your stripped wall socket cable to the AC side of the bridge rectifier.
The plug it in to charge. Will take 1-2 seconds to charge tops.
Turn power off at wall socket, disconnect bridge rectifier from capacitor bank.
To fire, connect the capacitor bank to the antenna. It will fire through the ring one way (and you're probably going to fire it backwards first time.)
Too slow to assemble? There's a faster, but last realistic, way.
There is another way to do it - its much lower power and less likely to work, but you could make it under a minute.
Use scissors on an extension cord to make a loop as per above, ends stripped, with positive and negatives joined together (so they function as a single conductor).
Power supplies in servers are hot swappable. Rip a power supply out while its live (ignore the warnings saying don't do this).
Disconnect the power from the wall (leaving the capacitors charged).
Connect your antenna to the 12V and ground output of the power supply.
This uses the charge of the capacitors still inside the power supply. If you only need a tiny amount of EMP, this may be enough.