What does "Maximum failsafe period has expired" mean?

It probably refers to the retry cycle that most mail transfer agents use in the face of error conditions that may be temporary (i.e. "can not contact server" is potentially a temporary problem due to network/DNS issues but "user does not exist" is pretty much considered to be a permanent state for the purpose of the current message).

What usually happens in these cases is that the message is dropped back on the queue to retry sending in a few minutes. After a couple of retries the mail server will start retrying every hour instead of every few minutes. After a few more it will start retrying every 24 hours, and after a few days it will bounce the message back. The amount of retries, and the periods between them, is not a defined standard: different mail servers will use different timings.

So if I were to get the "Maximum failsafe period has expired" message in a reply/bounce then I would assume the destination mail server is inaccessible.