It sounds like the compressor is either over heating or the unit is shutting off on a refrigerant pressure control ( either high pressure or low pressure ) .
Does the outdoor unit fan come on at all when it is trying to start ?
It sounds like you have a bad contactor in your outdoor unit. It looks like this one.
From what you describe it sounds like a contactor problem. If it were a compressor problem the fan should stay running the whole time that you hear the humming noise. So I'm discounting that it may be a compressor problem however it could be but it just doesn't sound like it right now from the symptoms, there is also another possible defective part which is the capacitor.
In your situation it will be worth having a service call to have it checked as there is a good possibility that you won't need a large item such as a compressor. A capacitor or a contactor are the two most inexpensive repairs that can be made on a unit.
No , I would kill it in case it is the compressor not being able to start because of the capacitor. There isn't much chance that it is going to start if either of those components have failed, or even if it is the compressor.