I'm basically trying to setup my own private pastebin where I can save html files on my private server to test and fool around - have some sort of textarea for the initial input, save the file, and after saving I'd like to be able to view all the files I saved.
I'm trying to write this in python, just wondering what the most practical way would be of storing the file(s) or the code? SQLite? Straight up flat files? 
One other thing I'm worried about is the uniqueness of the files, obviously I don't want conflicting filenames ( maybe save using 'title' and timestamp? ) - how should I structure it?
Plain files are definitely more effective. Save your database for more complex queries.
If you need some formatting to be done on files, such as highlighting the code properly, it is better to do it before you save the file with that code. That way you don't need to apply formatting every time the file is shown.
You definitely would need somehow ensure all file names are unique, but this task is trivial, since you can just check, if the file already exists on the disk and if it does, add some number to its name and check again and so on.
Don't store them all in one directory either, since filesystem can perform much worse if there are A LOT (~ 1 million) files in the single directory, so you can structure your storage like this:
FILE_DIR/YEAR/MONTH/FileID.html and store the "YEAR/MONTH/FileID" Part in the database as a unique ID for the file.
Of course, if you don't worry about performance (not many users, for example) you can just go with storing everything in the database, which is much easier to manage.
Storing files for testbin/pastebin in Python