If you've disconnected everything from the circuit there are only two causes I can think of - either the breaker is bad, or there is a legitimate arc occurring and the breaker is doing its job and possibly preventing a fire.
If you are comfortable opening your breaker panel, you could try replacing the breaker with a new breaker. If it still trips, I think that's a pretty definitive sign that there is an arc occurring. At this point, you probably want to call a licensed electrician to come diagnose and solve the issue. It could be a bad connection or a damaged wire.
From the comments posted above I understand....
You say that you measure a hot to neutral short at the electrical box for one particular circuit. On the other hand you say that no short is measured across hot to neutral at the outlet boxes.
This can mean one of two things.
(One) Something severed one or both of the wires between the outlet string and the breaker box and in the process caused the two wires to become simultaneously shorted on the breaker box side of the break.
(Two) The shorted pair from the breaker box is feeding a different circuit than you think it is.
For the first case you should really think about what has changed in your building between now and when the outlets used to still function properly. In most cases it is unlikely that wires spontaneously break and short together without there being some interposing disturbance of the wiring. This can give a clue of where to start looking. Did something get nailed or screwed to a wall someplace? Did a window get replaced?
I can think of one case where a wire break/short could happen and that is if there was a loose or high resistance connection that over time caused so much heat to be produced wherein the wires melted apart and also shorted due to the insulation burning off. This type of failure is most likely going to happen inside an electrical box itself where connections exist.
For the second case the obvious thing is that you need to guide your attention to the correct circuit. Note that I have seen many cases where the breaker function labels in the breaker box are incorrect. So double check the circuit carefully.
To aid in either case there is a simple technique where you can run a known good conductor from the destination point back over to the breaker box. Just lay it along the floor / ground. Then wire clip one end of it a wire in question at the destination and use the other end of the wire with your multimeter. This effectively makes one of the test leads of your multimeter long enough that you can look for continuity or opens between breaker box and destination without having to buy other types of test gear. This kind of testing should obviously only be done when all the power is turned off. Use a flash light or battery powered lamp if it gets too dark to see.
One other thing to think about....if not already being done....is that when testing for problems like this it is generally necessary to remove all connections from each wire run and test against wire ends that are free ended.
!
Best Answer
GFCI breakers are prone to fail earlier than traditional breakers. I would swap the breaker with another functioning GFCI breaker from the electrical panel to see if the problem follows the breaker or continues on the existing circuit.