OK, let's straighten this out!!!
IF YOU HAVE (like me) an older setup ---I use "dryer plugs" WHICH ARE NOT a neutral!!!! They are TWO wire plus ground, which is the green/ safety ground AND NOT NEUTRAL!!1
Therefore you CAN NOT (legally) use this for 120V (I do anyway LOL)
HOWEVER.........every 240V welder I've seen in the last 40 years had a FOUR WIRE PLUG!! This means it has TWO hot conductors plus the NEUTRAL plus the green/ safety GROUND
Each HOT is 120V to NEUTRAL. This means you have TWO 120V circuits in that plug
So IF you have a 4 terminal/ slot/ conductor receptacle, you could wire it for 120 EXCEPT FOR THE following:
You have a HUGE (30-40-50? amp) breaker feeding that circuit, and that is NO PROTECTION AT ALL for 120V outlets.
IF YOU ONLY want something temporary, I would get a small 2- breaker box and put som 15 or 20A breakers in there, and then wire it with a plug to fit the welder receptacle...........go to the box........and then either more cord to something like a couple of receptacles in a box, or mount the box right on the back or bottom of the breaker box.
WHAT YOU HAVE
GROUND/ green is for SAFETY, it hooks directly to the breaker box and the receptacle box, as well as to the green screw on the receptacles
You will need to devise some way of "stepping down" the big wire from the welder plug to fit the breakers. Wire nuts, or careful selection of breakers and your "welder cord" they might fit fine as is
One hot to each breaker, green to the box, and white / neutral gets connected to the white screws on the receptacles, so this will have to be clamped with a terminal/ wire nuts etc and passed on to the receptacles.