Why can't I update my OS? Whenever I try to download anything from the App Store it says "_____ can’t be installed on “Macintosh HD” because OS X version 10.11 or later is required."

And true enough when I check "About this Mac" I find I'm currently running "Yosemite 10.10.5 (14F2511)"

But when I click software update the app store checks and says "No Updates Available".

It also tells me I've recently (early September) installed updates for iTunes, the security update and Safari. 

What do I do?
Automatic software updates only do minor MacOS updates, not new versions. You need to download the newest MacOS version (High Sierra) from the App Store and install it.