When wiring a generator directly into your circuit box with a 30 amp breaker is 10 gauge wire big enough?

Generators should not be wired directly into your distribution panel. The generator should be wired through an approved transfer switch by a licenced electrician. To do otherwise would be sheer folly and leave yourself vulnerable to a void insurance policy on your dwelling should any mishaps occur because of the project not getting proper approval.

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The biggest danger to direct wiring a generator is that you can have the breaker feeding from the generator on at the same time that the main breaker for your primary service is also on. This feeds the generator power out of your building, to the utility pole, to any lines still connected to that utility service and possibly other customers or buildings. This supplies power to the utility lines when the repair workers may be under the impression they are not live.

On the other hand, if you direct wire a generator without a transfer switch, and fail to turn off your main breaker, when the primary service is restored you now have 2 sources of electrical power on the same lines and I cannot fathom the dangers involved.

But the direct answer to your question is, Yes, 10 gauge wire is sufficient for any 30 amp breaker provided you aren't running several hundred feet.