I can try to answer this.
In this case, "come" is part of the phrase "come close" which means "approach without actually touching/achieving." For example, a boat can "come close" to the shore or you can "come close" to hitting another car. You can also "come close" to getting a job or winning a contest.
Whatever the situation, to "come close" means getting close without achieving.
For the Italians to "come close" to total disaster, it means that they were approaching disaster but avoided it.
I hope that doesn't make it more confusing!