Genuino UNO 5V pin is reading 0.3V
There are a couple more places you can sample the 5V signal, see the photo here:



On the right, on the ICSP header, the pin marked VCC is also connected to the 5V signal. Try testing that.

(The other one is pin 2 of the ICSP interface for the USB chip, but the one on the right should be adequate for testing).

I don't know why you are mucking around with LEDs and resistors, just put your meter between Gnd and 5V.

I find sometimes that the hookup cable has failed, try testing that (or try with a different cable).