A Hard cap is the maximum (cap) level a building or structure can be upgraded, anything higher is not possible either because the upgrade button disappears at that level or because of the storage restrictions in the individual towns.
What this all means, is that any building level, which requires more than 2,562,500 of any type of resource to be get to that specific level even with a maxed out reduction buildings in the town, can not be expanded to that level and we will say that this is the "Hard cap level due to storage limits, without Ambrosia".
Ways to increase your storage beyond normal means
It is possible to double your storage capacity with 24 for 7 days, allowing you to upgrade building levels that need to use more than 2,562,500 of each resource but not more than 5,125,000 of each resource.
If a building requires more than 5,125,000 of any type of resource to be built to a specific level, even with reduction buildings in the town and with the use of ambrosia to double your storage capacity, can not be expanded to that level and we will say that this is the "Hard cap level due to storage limits, with Ambrosia".