October stores the content of your pages within the theme folders. This means that by switching themes, I'm also switching content for my website.

Other CMS's I've used such as WordPress retain the content completely separate from the themes, so I can switch themes and retain the same content and pages for the site without rebuilding or manually duplicating the pages.

How can I do this within October so that my content is separate from the theme? I've noticed that the blog plugin stores the content in the database and so that data is separate and portable. Would modifying the blog and replacing the CMS functionality with this be the best way to do this?

You have plenty options for that:

1. Using native features: keep all your content blocks in the Content section of CMS and just copy them from one theme to another
2. Using tricks (bad, only for texts): install http://octobercms.com/plugin/rainlab-translate plugin and give your snippets some id (eg. {{ 'snippet_code'|_ }}, and then fill them in Translate section.
3. Using plugins:
   • http://octobercms.com/plugin/clarknguyen-staticblock [paid]
   • http://octobercms.com/plugin/buzzwordcompliant-dbcontent [free]
4. Using brain: make your own plugin using awesome Builder plugin http://octobercms.com/plugin/rainlab-builder and customize however you want!