One of the original applications to apply the concept of no-nonsense, distraction free, full screen writing was WriteRoom for Mac OS X. The name spawned a number of clones, all similar, but often written in different languages. I refer to them as The Rooms and summarize them here because every editor is near identical. They all share the same full screen appearance, some have customizable background and text color options, and most perform the same duties: simple, plain text editing.
The most downloaded for Linux is TextRoom. TextRoom has many of the features of a modern word processor like LibreOffice's Writer, but maintains a minimalist interface. Many users like TextRoom because it's a cross between a fully stripped down interface with no formatting options and a fully populated, distraction rampant word processor.
JDarkRoom is another popular spin-off of WriteRoom with a hyper-minimalist interface. It has a goal feature, where you can input the word count you wish to reach and JDarkRoom will inform you when you have reached your goal for the day. As the "J" suggests, JDarkRoom was written in Java.
PyRoom is the final WriteRoom spin here, but certainly not the last one available. PyRoom attempts to be even further out of your way and minimalist than JDarkRoom and, as the "Py" suggests, is written in Python.
Since all of these WriteRoom alternatives are open source, they are free to use and modify as you need. The standard application will be good for those looking for a functional, full screen, distraction free writing environment though.