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AWG 2/0 copper wire.
As a service entrance wire you need AWG # 3/0 gauge copper.
Yes, they are the same size wire.
A 3/0 copper conductor with an insulation factor of 90 degrees C is rated at 210 amps.
A 200 amp service panel will require a # 4 bare copper ground wire.
This is a voltage drop question. To give an answer the voltage must be stated.
That certainly is a lot!! It sounds like you will need to install a 200 Amp subpanel in your barn. This means that your main power box will have to have room for a 2-pole 200 Amp breaker, in addition to whatever other main breakers that may be in it. As far as wiring: normally you could go with 3/0 copper, or 4/0 aluminum. However, with a 300-foot run, I would size up my wire by at least one. 4/0 for copper, or 250kcmil for aluminum MINIMUM. You only want a maximum of a 5% voltage drop from the main breaker to the barn's subpanel. Using the smaller wire size will probably give you a drastic voltage drop, causing tools to not work properly. Be certain the wire gets buried 18-24 inches below the ground.
Service wire required is AWG # 3/0 copper.
A 3/0 copper conductor with an insulation factor of 75 or 90 degrees C is rated at 200 and 210 amps respectively.
3/0 wire 3/0 wire
208 volts is not a common single phase voltage but you could just size your breaker to the capabilities of the conductor. 208 is normally a phase to phase voltage <<>> Many 230 volt motors can be connected to lower voltage supplies. These motors can be connected to supplies of 208 and 200 volts. The code requirement is that the 230 volt full load amperage of the motor be increased by 10% and 15% respectively for wire sizing and breaker selection. To answer the question you will need a two pole 15 amp breaker.
It is very likely that the 120A current you quote is the output current. Check your Welder Box and look at the input current. It is likely around 30 Amps. To wire a 30A circuit you will need 10 gauge wire and it is likely 3 conductots + ground. You will need a 230V Breaker. Connect the black and red wires from the 10 gauge cable to the breaker, white wire to neutral and ground to ground.