You can always get the app's APK and install it. You can sometimes get them through Aptoide or any other website, but you don't always find the latest version, or you may not find it at all.
There are 2 ways to get the APK from the Play Store:
1- The first way is APK Downloader. Without any downloads, go to goo.gl/l15JdV and paste the link or package name of the app and it will download it.
For example, if you want to download Facebook .apk, you can either write the link: goo.gl/6iQMVN , or the package name: com.facebook.katana
2- The second way is using Real APK Leecher. I attached it to this post. It is a 3MB java app that is like APK Downloader, and works the same way.
PS: None of these apps are by me, and none of them can download paid apps.