Can I share a 240V outlet between two devices?
I think for what you want to do:


temporary short term solution  
you guarantee you will never operate both the dryer and welder at the same time


You could consider building a 220V 30A power strip.
Without knowing your welder, I'll list parts from Home Depot that might work but you have to confirm their suitability for your use. 

Please review this URL to determine your plug NEMA type. It will probably be NEMA 10-30 or 14-30. Make sure the plugs, receptacles and dryer cord all match.

Note this is for standard 30A dryer. If you need 50A, you will need different receptacles and cord. Probably a range cord.
But I assume that since you are currently plugging your welder into your dryer receptacle, 30A will be sufficient.


Raco 2-Gang 42 cu. in. Square Boxes 
Model # 8257 Internet # 202058366
You should screw this to the wall if possible for safety, as that will reduce stress on the dryer cord.


6 ft.10/4 4-Wire Black Dryer Cord
Model # AW20009 Internet # 100672788 Store SKU # 601004


(two) Leviton 30-Amp 2-Pole Flush-Mount Outlet
Model # R50-05207-000 Store SKU # 621336


wire, wire nuts, box cover, cord clamp etc.
If your box can be mounted close enough to the existing outlet, you can cut the excess off your dryer or range cord and use that excess wire to connect the two new outlets. This is just a very minor optimization and you might figure it out as you assembled the box but this way, you'll know before you go shopping.