The A6000 has an OLED eyepiece as many cameras do these days. These viewports have sensors to detect when your eye gets close so it can activate. When it works well, this is a great feature because it turns off power to the large, rear LCD. But when it's too sensitive, it can be maddening. (You can check out reviews on Amazon.)
It appears to be 4.5" from the eyepiece. This is just too close for me. You'll know if it's a problem for you.
The Sony A6000 is a truly awesome camera for the money and this small problem is easy to fix.
Step 1: Fix It!
You just need a small piece of electrical tap, a razor to cut it and some tweezers. The photo showing the dismantlement is just to help you understand how it works. You will NOT need to take apart anything.
The way the eyepiece works is it has a small infrared LED (light source) in the lower half and a sensor to pick up reflected light in the upper half.
The width of electrical tape sliver is 130mm (very tiny).
The "cup" or "hood" for the eyepiece just pulls off.
You want to position it just below the midpoint of the shinny black part of the eyepiece. You are trying to cover part of the LED light source on the lower half without blocking the sensor on the upper half.
If you get it in the wrong position, the sensor won't trigger. It's easy to test and tweak until you get it just right.
Step 2: The Result
As you can see from the video, the distance after the fix is a reasonable 1" - 2" from the eyepiece.
You can experiment with the width and placement of the electrical tape to fine tune the behavior.
The real fix
Sony should add a setting to adjust the brightness of the LED lightsource.