Currently my GUI does not work & I can only use the CLI to operate my distro.
How can I mount a portable drive, copy and paste all my files from my broken Distro onto that portable drive so I can save all my work before I reinstall fedora onto my system?
It might be easier for you to use a live session.
A live media can be created on another machine.
Will I be able to put it on my machine with the broken distro, and grab my files off of it?
Recovery scenarios are a typical use case for live media.
Since you want to reinstall the system, create it anyway.
I have tried it, and it seems that there is not enough space on a 32GB usb to put fedora on, even though I erased the USB of all of its data before putting the iso onto it. Is 32GB somehow not enough to make Fedora a live session?
You can create a live media by following the documentation:
Besides, I guess you could even reinstall the system without overwriting existing data. You should make a search for the detailed instructions but I suppose all you have to do is to set the “partitioning/format” step of the installer to NOT make changes on the /home directory.
It may worth a try AFTER you have saved your data on the external drive, just in case.
I don’t recommend using Fedora for this as there’s no Live session persistence by-default (no data is saved from the live session after reboot).
My personal favorite is using openSUSE Tumbleweed’s XFCE image for anything needing done from Live sessions; it has persistence, and it’s light. The image is about 900MB so that’s a good 30GB for anything else 
I would also keep another drive around for installing distros from; erasing the openSUSE live image from that drive, copying another OS installer image to it, and keeping the persistence data could probably be done, but it’s much easier with another drive dedicated to it.
Yes, I thought the idea was to boot from the “live” USB and connect ANOTHER external drive to copy files, not the same USB because of course without persistence it will not retain anything. Good catch.