We all know what an hour is sixty minutes, each minutes consisting of sixty seconds. There are twenty-four hours in a day, and days are further collected into weeks and months, which then collect to become years. We also know that the word "hour" in English doesn't always mean exactly sixty minutes. It can mean a particular moment, as in "the hour of decision," or it can mean a general time, as in "the family hour." Those hours are extremely flexible, as it happens.
"Sha'ah" means "hour," and it has been used for that meaning since a very long time before we had mechanical devices to divide the day into the artificial ticks of the clock that are so familiar to us now. In Talmudic times, and hour was one-twelfth of the measure of sunlight in any given day, which means that by those standards an hour could be clocked at somewhere between about 53 and 67 minutes in Israel. (In certain latitudes, by that standard, an hour could be more than two weeks long!) Though sha'ah has come to have a more precise meaning in Modern Hebrew, the less exacting definitions of "hour" also apply to "sha'ah."
Sha'ah also can mean a particular moment in the very first Mishnah at the beginning of Tractate B'rakhot there is a reference to the "sha'ah" that the priests sit down to eat the offerings. It can mean a general time when we recite Avinu Malkeinu we ask that this be "sh'at rachamim," an hour of compassion from God. The instant and length of each of these kinds of hours, of each sha'ah are vague.
So if sha'ah is so variable and imprecise, how are we to understand it in prayer? I would suggest that it carries with it the meaning of moment not so much a particular moment in time, but a momentous time, something worth noting. If, as I previously taught, the Hebrew word "eit" means the very expansive sense of "phase of life," then sha'ah means the very specific moment of which we take note "moment" in both senses.
For God to consider us "b'khol eit u'v'khol sha'ah" is for God to consider us in every phase of life and in every moment, to be concerned for us over the long haul and at each point along the way.