The App Store and iTunes apps both use Safari to load web content, and sometimes resetting Safari helps if they are crashing.
First, try the following:
1. Open Safari, press Safari on the menu bar and click Reset Safari.
2. Uncheck all the options except, Remove All Website Data and Close All Safari Windows.
3. Press Reset.
This will clear the Safari cache, and may help.
If that doesnt work try booting in Safe Mode:
1. Turn your computer off.
2. While you turn the computer back on, immediately hold the Shift key until you see an Apple logo with the loading bar underneath.
3. It will load your login page with your user, log in from here.
4. Test the issue, and reboot.
If all else fails, try clearing your system cache:
Pull the App Store plist
1. Open Finder and select Go on the menu bar.
2. Hold the Option key and select Library.
3. Go to the Preferences folder that you see in there, and navigate to the com.apple.appstore.plist file andcom.apple.appstore.plist.lockfile and drag them both to the desktop.
4. Reboot and test the issue again.
Let me know if you are able to get into the App store again?
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