Why is my basic DAC incorrect?
There are two issues here. It is true that for the kind of DAC you constructed:
$$V_{MAX} = \frac{2^N-1}{2^N}\times V_{REF}$$
where \$N=8\$ and \$V_{REF} = 5.000V\$ in your case. So you will never get exactly 5V out...the difference is always the voltage equal to a change in the LSB, which is also called the resolution of the converter.
The other issue is that you have connected the R2R string incorrectly. You need to use SPDT switches instead of SPST switches, so that the end of each of the 2R resistors is connected either to \$V_{REF}\$ or to ground.