I use http://paste.ubuntu.com/ on AskUbuntu and on almost all other SE sites, (SU, U&L etc...). Is it okay to do this ? or should it be used only on AskUbuntu?
What are the restrictions for using Ubuntu Paste-bin ?
None whatsoever. It's a free service and you can use it where you want. Note, however, that in most cases it is better to include whatever you need to reference in the answer itself.
But yes, if you need to include something that's longer than the character limit, use whatever service you prefer. Ubuntu Pastebin is not an Ask Ubuntu thing. It's an Ubuntu thing and can be used anywhere.