Is this the common fuel pump relay problem? Do I need to just swap out fuses or buy/clean/fix the relay that's under the fuse box? It eventually turns off after car shuts down but it definitely runs for a good five minutes all by itself.

when you shut the key off you are supposed to be cutting all power to the vehicle...except lights. For some reason power is still being supplied to the relay or pump itself. Sounds like a possible short to me inside the relay...maybe change it out but that doesn't explain why power is still getting to the relay...hmmm I'll get back to this one

If I were you I would at least replace the fuel pump relay switch. Yes it is a common problem in older Explorers. It is a simple and cheap ($15) repair. Most people that experience the problem think it is a battery/alturnator problem, because it runs doen the battery. It is also a good idea to replace the EEO relay beside the fuel pump relay. I would also use motocraft switches. I got one at advance and it was either defective or just the wrong switch, it made my voltage go crazy. Use a little wd-40 as a lubricant on the plastic parts to make them slide easer.

i dont think you want to use wd40 on plasitic anything, it can and will dry the plastic out and cuase ti to be brittle...

I agree it is most common for the fuel pump relay to be the issue.

Relays use a magnetic coil to pull the contacts (Power & Load) together when a trigger voltage is applied (this is the clicking noise they make). When the trigger voltage is shut off springs force the contacts apart. Relays tend to fail when their springs get old and worn. This prevents the contacts from immediately separating when the trigger voltage is no longer present. As in your case the springs are able to force the contacts apart after several minutes.

OK use pam (vegatable oil). Brittle plastic on a 91 Explorer, how could that be. All the plastic on my X is brittle.

when you shut the key off you are supposed to be cutting all power to the vehicle...except lights. For some reason power is still being supplied to the relay or pump itself. Sounds like a possible short to me inside the relay...maybe change it out but that doesn't explain why power is still getting to the relay...hmmm I'll get back to this one
The fuel pump relay has a permanent power supply (key on or key off) at one of it's four leads (yellow wire I believe). This is on the fuel pump circuit side of the relay. If the relay gets stuck closed, then it will run the fuel pump steady, even with the key off.

Fuel pump relay?? I believe it's the green one.

The agtermarket ones are all the same. Ford has different colors with plastic bumps on them so they are Keyed.

The fuel pump relay has a permanent power supply (key on or key off) at one of it's four leads (yellow wire I believe). This is on the fuel pump circuit side of the relay. If the relay gets stuck closed, then it will run the fuel pump steady, even with the key off.But with the key in the off position power should be cut to the other side of the relay allowing the contacts to open...sorry for the misconfusion....bottom line is a problem with the relay. Thanks for pointing out the confusion Shorty my mistype.

So I'm clear on this, this is not the fuse in the upper box but the relay which is under the box or to the side of it?

It is the fuel pump relay located in the power distribution box near the stock air box. Both the relays and fuses are accessed by opening its cover. The components should be clearly marked.