Yeah, if you're using a Chromebook then you'll need to find something online/cloud-based to use rather than something installable like Open/LibreOffice (which I would otherwise recommend).
Google Drive is all right; I used it pretty extensively a long time ago, although I've since been kind of annoyed by some of the interface changes they've made (in particular with regard to the formatting; I don't like what it does when you try and paste it into a forum's WYSIWYG mode reply box, and I end up having to clean out a bunch of unnecessary formatting BBCode tags) and don't use it much for writing fiction anymore. Issues aside it probably is the best free online word processor available, though, and it may offer some integration with your Chromebook at that since it is Google, so it's probably your best bet.
Zoho Docs is similar and still fairly popular iirc, so you might want to look at that as well. Personally I don't think it's as good, but some people like it, so...
If you don't mind manually writing out the BBCode tags/using something simple like a Notepad clone as opposed to something more akin to Word, you can also look at things like
Writebox. (Writebox in particular also has a Chrome app that basically does the same thing, if you'd rather use that than visit the url.) Writebox requires a Dropbox (or Google Drive) account if you want to save documents more permanently, and as stated above it doesn't have most of the features of a full-blown word processor the way Drive does, but it's a good distraction-free writing environment for when you need to concentrate, and hey, if there're no formatting options then that also means there won't be any formatting cruft to clean up later when you're ready to post it someplace like Serebii.
Really, a search for "online word processor" or "cloud-based word processor" or similar should turn up alternatives in case you want to try something other than/in addition to Google Drive.