If you follow this guide step-by-step, your old Samsung Galaxy S III, Nexus 7, or HTC One X will boot into a functioning Google Play Store once again. Enjoy the nostalgia, but do not rely on it for security-sensitive tasks.
If you have tried everything and the Play Store still crashes, accept that the Google servers have permanently turned their back on API level 16. Wipe the device, install a lightweight launcher, and use APK mirrors manually from your computer.