When I choose one option, the other one becomes disabled. Is there any way around this?

Could you please be more detailed? Is this in a theme or plugin?

There are many, many ways to provide a combination of static and dynamic content. I use custom page templates to output dynamic content and the_content (the content in the pages page editor).

Other examples are shortcodes and many varying themes and plugins.

First of all, you need to choose static option in the WP settings. Then, on that page you can do anything you want. For instance, I installed a plugin which makes my home page dynamic. I mean that each time I publish something, it is shown on the home page but in the way I want.

You create such page, create a navigation menu for this page to help your visitors to find it, and that’s it.

I guess, what I am trying to say is that a WordPress page has to be either static or dynamic. It can not be both.