Is there a way to store a view in a phtml file and then render it as part of another view? For example, if I have a menu or footer that is common to all of the pages on my site and I want to keep it in one phtml file so that I can change it in one place and it will change everywhere. I can't find anything about doing this in the documentation.

Sure, you just need to call the render method in your view script file like this:
Although, I'd advise that you look into the Zend_Layout library as this implements the 2 step design methodology. You'll quickly find that you'd prefer to do things this way if you start to realise that you're including headers and footers in every view in your application.

More info here: Zend Framework: Documentation

Also, outside of Zend_Layout is the Partial helper for Zend_View. So if you need other repeatable elements that isn't apart of your site wide template (layout), then partials are the way to go: Zend Framework: Documentation: Zend_View (Scroll down to this section: 41.4.1.2. Partial Helper)