Q:

Can't balance differential amplifier circuit

I'm trying to build a circuit which involves an  Op Amp in differential amplifier configuration. 
 However I've been having difficulty balancing my output. My understanding of the differential amplifier is that if configured such that R1 = R2 = Rf = Rg then Vout should be about 0 volts except I'm always getting about 0.5-0.8 volts.
I've currently set it up so that there are 5 volts going directly into R1 and R2 with all my resistors equal to 10K ohms and still I'm getting about 0.63 volts. These are diagrams of my circuit I made:

Please also ignore the dashed lines.
This is what I was trying to build:

Edit: Original Circuit

A:

The LM358 (like other op-amps in that similar range) has the ability to pull-down its output to close to 0 volts because it uses a 50 uA current sink on the output. Without that current sink, the minimum output voltage would be about 0.6 volts above the negative rail (0 volts in your example): -

And here comes the problem; your feedback resistor is acting like a pull-up resistor to 2.5 volts (set by R2 and Rg) and this negates the effectiveness of the internal 50 uA current sink because Rf will try and inject 250 uA into the output pin thus lifting it up above 0 volts.

