There are many ways, of course, but I start with one which is simple to explain: it just requires some command line magic. I assume that there is internet connection (check it for example with ping -c 5 google.com. Also, I assume that /my/file is the location of file you would like to submit, and JohnDoe is your askubuntu nickname.
The simplest of it all, but you need to install pastebinit first:
sudo apt-get install pastebinit
pastebinit -i /my/file
Submit using curl. For that, curl must be installed.
sudo apt-get install curl
curl -v --data-urlencode "content@/my/filename" -d "poster=JohnDoe" -d "syntax=text" http://paste.ubuntu.com
You will see some output, with one of the lines looking more or less like that:
< Location: http://paste.ubuntu.com/123456789/
This is the link you need to share.
There is a number of text-only web browsers, for example w3m. Run these commands:
sudo apt-get install w3m
w3m http://paste.ubuntu.com
You move around with a cursor. When you come to the text area and press "Enter", w3m will ask you which editor to use; choose nano. Use CtrlR to read in the file.
Your USB sticks still work. Insert an USB stick and type mount to see at what location it has been mounted. Say, you see that it is /media/MyStick. You can copy the file using command line
cp /my/file /media/MyStick
