The bare minimum required to install applications from the Play Store is the Google Package Installer and Google Play Services. Google and Android are two separate entities, and although CM is based on Android it does not tie in the Google services, but it is compatible with them. You do not need Google to use CM, and there are alternatives, but to use the Play Store you MUST install the minimal services listed previously.
OpenGapps has the smallest package available to be able to access the Play Store and it contains only the absolute minimum required to access and install apps from the Play Store called a "pico" package.
Of course, free apps could be found in a reputable APK mirror site, like APKMirror, but this is not a valid location to find paid apps. Be aware that some features of apps distributed via the Play Store require Google Play services in order to function, so in some cases without it installed some feature may not work, and it is possible that entire apps would not function, although I do not know of any examples at the moment.
Other potential sources for apps would include the Amazon App Store, GetJar, MoboGenie, SlideMe, and F-Droid, which are all safe, viable alternatives to the Google Play Store.
MakeUseOf did a respectable article earlier this year entitled "Here’s How You Can Use Android But Ditch Google", a read through this article may help your understanding of getting along without Google.