From a mathematical/algorithmic point of view, you could define a good explanation as a sort of compression process: on one hand, you have observations, data to explain, totalling a certain number of bits. On the other hand, you have a process or algorithm that can generate these observations, and if that process can be described in less bits than the original observations, then you have a "good explanation". For example, our current theories for the laws of physics are excellent explanations, because they can explain a virtually infinite number of real observations from finite information.
On the other hand, if the observations are truly random, then in general no shorter process can produce them, so there can be no good explanation for them. And the interesting thing is that if every good explanation compresses the original observations at least one bit further (otherwise they would not be good), there must be a point where the result is as short as it could possibly be. At this point, the series of good explanations would have to end (although I believe that it is undecidable to know when the end is reached).
It is also always possible for something that has a good explanation to actually be a brute fact, like the idea that the Earth was created with the appearance of old age: the good explanation would be that it aged, but the truth would be that it didn't.
I don’t think the length of the explanation is very relevant, and finding shorter explanations doesn’t seem like a primary concern. Explanations should be judged on what problems they solve and how well they stand up to criticism and competing explanations. And since I don’t think any explanation can be “final” or “100% true” or “guaranteed” or anything like that, the notion of a shortest possible explanation doesn’t even make much sense.
On the other hand, if the observations are truly random, then in general no shorter process can produce them, so there can be no good explanation for them. And the interesting thing is that if every good explanation compresses the original observations at least one bit further (otherwise they would not be good), there must be a point where the result is as short as it could possibly be. At this point, the series of good explanations would have to end (although I believe that it is undecidable to know when the end is reached).
It is also always possible for something that has a good explanation to actually be a brute fact, like the idea that the Earth was created with the appearance of old age: the good explanation would be that it aged, but the truth would be that it didn't.