What size is my electric service?

You have a 60 amp service. A 60 amp service can be connected to a 200 amp meter box. The 600 volt rating means nothing, your home service is 240 volts. I would consider installing a new 100 amp circuit breaker panel in place of the main fuse box with the 60 amp fuses in it. You will need new wires from the meter to the new panel if you do that.