PasteBay is described as 'Free uncensored text hosting'. There are more than 10 alternatives to PasteBay, not only websites but also apps for a variety of platforms, including Windows, Linux, Mac and Android. The best alternative is Pastebin.com, which is free. Other great sites and apps similar to PasteBay are GitHub Gist (Free), hastebin (Free, Open Source), Ideone (Free) and 0bin.net (Free, Open Source).
The website is mainly used by programmers to store pieces of sources code or configuration information, but anyone is more than welcome to paste any type of text. The idea behind the site is to make it more convenient for people to share large amounts of text online.
0bin is client side encrypted pastebin that can run without a database. It allows anybody to host a pastebin while welcoming any type of content to be pasted in it. The idea is that one can (probably...) not be legally entitled to moderate the pastebin content (...
Very simple webpage with a simple textbox, a checkbox for selecting one of several languages and an option to make it private and to run the code or not. Has a little code sharing and references as well.