Making a pure R-2R resistor ladder DAC with adjustable offset
Well you can certainly get op-amps that do work well from a 5V rail but ignoring that part of the question and concentrating on the two 0V points on the first circuit, if these were at 1V then there would be no problem. However, the outputs from the MCU (shown as switches connected to either 0 volts or positive rail) cannot be disentangled from 0V.

But, if you used a bunch of analoge switches (as you have basically drawn in the diagram) this would work fine. The two triangular shaped ground points could connect to any abitrarily low voltage (such as 2V or 1V) and the range of your DAC would stretch from whatever voltage you choose upon for the "ground" up to whatever voltage you choose for the other supply to the analogue switches.

There are suitable parrts that run from 5V and give very low impedance for "on" and beyod-the-rails capability. I'll try and find one to recommend.

Here's a single device (one-eigth of the way) that does SPDT but I'm sure with a bit of searching a quad device could be found (half way there): -