810 amps at 3.7 volts is 3000 watts, which is 150 amps at 20 volts, plenty for arc-welding; different 18650 lithium-ion (Li-ion) cells are rated at 10–35 amps, with most being 20 or 30 amps in theory; being cautious and only expecting 5–10 amps out of each cell, you’d need 80–160 cells, quite a reasonable quantity really. I mean a 9×9 array of cells would be 81 cells.
Charge and discharge rates are rated in “C”; 1C is a current that would yield the battery’s rated capacity over one hour. 2C and 3C are typical charge rates, but discharge rates range from 2C to 15C, depending on application. Typical capacities range from 1000 to 3000 milliamp hours.
Dimensions are 18.5 mm diameter, 65 mm length, 47 g, so an 11×11 array of 2000 mAh 15C 3.7V cells would weigh 5687 g and deliver 3.6 kA. So uh I guess you could probably go with a quarter of that: a 6×6 array of 2000 mAh 15C 3.7V cells, or using a hexagonal array⁰, 37 cells in a 7-across-the-corners hexagon (130 mm). This weighs 1700 g and can deliver 1100 amps at 3.7 volts, 4100 watts. An appropriate output circuit could convert this to 70 amps at striking voltage of 60 volts, then 200 amps at arc-sustaining voltage of 20 volts.
⁰ N.concatenate(([1,], 6 * N.arange(1, 10))).cumsum() # [1, 7, 19, 37, 61, 91, 127, 169, 217, 271]
These cells cost about US$10 each, so the whole battery pack would cost about US$370.