How to do it:
Disable ALL your GAME plugins through the PSP Recovery Menu, else the PSP won't have enough available RAM to run DOSBox + the game (it will need as much as it can have).
Install Fallout on Windows or DOS wherever you want. Go for the latest version if you have it (GOG), and use Max install.
Copy the Fallout folder to somewhere in your PSP memory stick (I have it like root:\DOSBox_C\Fallout).
Download this ZIP (v5).
Inside "For the Fallout folder", choose the correct folder (PSP 100X (Fat) vs any other PSP type) and copy everything inside that chosen folder into the Fallout folder.
If by chance you don't have the DOS executable file, get it from Fallout Fixt mod (tick the option to include the MS-DOS executable file) or from TeamX patch 1.2. In Fixt it's called FALLOUTD, in TeamX's patch it's FALLOUT.
Use DOS/32A instead of DOS4GW if you'd like - it's an improvement and it's faster. For that, just download it from https://sourceforge.net/projects/dos32a, rename it to dos4gw, and delete or rename the old one.
Copy the DOSBox 0.72 folder (a HUGE thanks to u/captchalove for updating DOSBox for PSP) to PSP\GAME and configure the path to your Fallout *DOS* file on dosbox.conf if you didn't use DOSBox_C\Fallout\FALLOUT.EXE as the file path like I did (the same that comes in the dosbox.conf file I give).
Download the F1DP mod, install it (read its README to know how to install it), and configure it. Use Ctrl+F to search "delay" and set all delays to 0 (CombatPanelAnimDelay, DialogPanelAnimDelay, and PipboyTimeAnimDelay). Also set FADE_TIME_MODIFIER to 0 and SkipOpeningMovies to 1. This will get the game as fast as it can be when it starts up, and hopefully too when playing (because of the animation delays).
NOTE for the PSP 100X: is has only 32 MB of RAM compared to the 64 MB of all the others. That means DOSBox can only have 14 MB to work (thank you u/ClassyTheCooller for the testing). Fallout's minimum requirements are 16 MB of RAM... So to do magic, one not-so-cool thing had to be done: fully disable the sound ahah. That's apparently what it takes to run it on the Fat.
Fallout Fixt also works on the DOS version. If you'd like to use it, check here: https://www.reddit.com/r/classicfallout/comments/sjpw0s. Note though: I never tested Fixt on the PSP. It may or may not have enough memory to run it, especially the Fat (or maybe no additional memory is required. No idea. If anyone tries and wants to say, feel free and I'll update this).