In HTML, tags and entities aren't parsed within <script> tags, and </ immediately ends the tag. Thus,
<script><b>fun & things</
will give you a script tag with the exact contents <b>fun & things.
If you're including JSON and you want to include the characters </ in your script, then you can replace it with <\/ because the only place for those characters to appear is in a string, and \/ is an escape sequence that turns into a single forward slash.
However, if you're not using JavaScript, then this trick doesn't work. In my case specifically I'm trying to insert a <script type="math/tex"> into the source so that MathJax will process it. Is there a way to escape </ in the original HTML source? (I don't have a particular need for </ but I'm writing a generic tool and want to make it possible to use any text.)
(It's possible to create the script tag in JavaScript and populate its innerText, but I'm working with the raw HTML so I can't do that.)
In HTML, as opposite to XHTML, the content of a script element is processed as plain text except for the occurrence of an end tag, so that </ ends processing and must, in conforming documents, start the end tag </script>. There is no general mechanism to avoid this. Any methods that circumvent this feature are unavoidably dependent on the “language” used inside the element. The word “language” is in quotes here, because the content can be just about anything, as long as your code can parse and process it.
So: no general mechanism, but for content other than JavaScript or some of the few other client-side scripting languages recognized by some browsers, you can make your own rules.