Use the APKMirror link above to save the file to your device.
This is because the Google Play Store is not a standalone app; it is a client that relies on Google Play Services. The "backend" of the Play Store has evolved. The servers Google runs today speak a different protocol than they did in 2013. The API endpoints used by Version 4.1.16 to fetch app lists or purchase data have been deprecated or altered. play store version 416 download link
Distributing the Play Store via third-party links exists in a gray area. While Google allows the installation of their apps on Android devices, the company does not officially sanction the redistribution of their proprietary code via third-party sites. Use the APKMirror link above to save the file to your device