If one side is dead out at the shed (and all the fuses, supply and load side BOTH are all good), the problem could well be a loose or burned connection at EITHER the SUPPLY (like home or barn etc) orrrrrrrrr the LOAD box at the shed.
Or heck maybe one of the feeder wires out to the shed opened up?????
Id inspect all the lugs and cables and clamps and terminals to insure the screws/bolts are good n tight and the wire or terminals arent all burned and carboned.
Rememember, the problem could be back at the box that feeds the shed instead of at the shed itself!!!!!!!!!
Welding draws high current so I bet you will find a loosed or burned or carboned up connection/terminal/clamp at the supply or load boxes (assuming all fuzes are okay)
NOTE a voltmeter might not show all the problem (unless its totally broke open) since they are such high resistance and draw so little current, its that high current draw welder that really tests the circuit.
PS if youre talking 120/240 volt single pahse three wire, there should be 120 volts from EITHER hot leg (L1 or L2) to Neutral but 240 volts from Leg to Leg. Since you mentioned "shed" and if its been wired for years theres a darn good chance there were only three wires ran out there, two Hots and the Neutral. Regardless, that shed electrical service (older 3 or newer 4 wire) should still have a Ground!!!