I hear you on this; I'm in the situation where neither my small shop nor my garage has 220-240V available. Also, I don't know if my current service will allow for another such circuit - in any case, I'd have to likely add an auxiliary panel to accommodate the extra breaker(s) needed, as my main panel is full.
But I do have one possibility - and maybe you do, too?
From the door to my garage that leads into the house, the laundry room is just around the corner. In that room, I have both the dryer and water heater, on separate circuits. So in theory, I have the voltage needed.
Now, the water heater is hard-wired, so nothing I can really do there (well, I could - but I won't). The dryer, on the other hand, is plugged into a 220V outlet (as they usually are).
So - what I have considered doing is making an extension cord; one with a plug for the dryer, and on the other end a socket for the welder. In case you aren't aware, the plug on a welder for 220VAC is different than the plug on a dryer. And technically, you're not supposed to use a welder on such a socket (there's something in the code that forbids it).
But people have been making and using such extension cords for almost the entire time such welders have been available, and I have not heard of any reasons against them. You can even purchase such cords thru some places, so they're even legal to sell, purchase, and own. It may be something having to do with the strain a welder puts on the plug and extension; that the extra connection can heat up too much?
Anyhow, maybe you could do something like this; get an IR thermometer and monitor the temperature of parts as you weld and use the circuit; see if there is any issues in that regard.
Another thing I've thought about doing, since the wall where the socket for the dryer is shared with the garage, is mounting a junction box, bringing in the 220, then tapping off it inside the junction box to a new socket and the old dryer socket. Then I could plug right into the socket in the garage without needing a custom "extension cord", so long as I unplug the dryer (maybe add a cover to both as well that can be locked to prevent both being used at the same time).
Something tells me, though, that such a setup would not be up to code - but I don't know for certain.