Is highly uncertain...
Let's start with the amount of carbon that has to be emitted to produce and deliver a kilogram of concrete. According to the Inventory of Carbon and Energy (ICE v1.6a), the embodied energy of concrete averages 2.9 MJ/Kg, but the standard deviation is 8.7 MJ/Kg, and the range in 122 studies worldwide goes from 0.07 to 92 MJ/Kg. So it all depends, and the impact of any concrete project is highly variable based on luck of the draw: how energy efficient the local cement plants, limestone quarries, gravel mines, etc are, how close they are to the building site and each other, and so on.
In any case, ICE reports a central value of 0.035 kg of carbon emissions per kg of concrete. Let's work with that for a minute.
Consider again the back-of-the-envelope house we were discussing the other day, which had interior dimensions of 45' x 29'. Let's suppose a fairly conventional kind of construction with a 4" slab on grade, and a 12" wide exterior footing wall that extends down 2'. That requires 27.35 cubic yards of concrete, which at a density of about 4000 lb/yard weighs around 50 metric tonnes. Thus the carbon emitted to produce that much concrete is 1.75 tonnes. A more complex plan, imperfections in site preparation, interior footings, etc, would probably raise the total by a few tens of percent.
This is substantially smaller than the 6.5 tonnes of carbon captured in straw bale walls. However, if the building department made us put in 24" wide footings, we would go up to 2.5 tonnes of emitted carbon. If we wanted a basement wall that thick, instead of a slab on grade, we would get up to 6.75 tonnes of emitted carbon. So we probably don't want to do it that way (and there's no reason to).
There are a variety of more creative alternatives, ranging from concrete blocks, rubble trench foundations, grade beams, earthbags, tires with rammed earth in, etc.
We are still neglecting lots of important things in this building (rebar in the concrete, carbon captured in the lumber elements, embodied energy of plumbing/wiring, etc), but those will have to wait for another day.