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Xperia 8: Phone asks to insert memory card when I attempt to take photo.
Xperia 8?
For Xperia Z3 Compact:
If you don't have an SD card,
maybe you have specified "Data storage: SD card" in the Camera app.
To fix that:
Start the camera app and press the three dots in the lower left area.
Press the third tab (Settings) in the menu showing up.
In the bottom of this menu, there is a "Data storage" option. Try to change it to "Internal storage".
If you have an SD card,
then you would need to troubleshoot that part a bit more 
You need a SD card to store pictures on the Xperia X8