The answer to your question is that "were" is incorrect and unacceptable here. The subject is singular and consequently it needs a singular verb. It is not acceptable to use a plural verb here even in casual speech, which isn't to say it wouldn't be understood, it would just not be grammatically correct.
For sure there are alternative plural subjects you could use. For example:
Our years of greatest prosperity were those eleven when ...
Our times of greatest prosperity were those eleven years when...
There are cases when it is less clear. For example: "The football team is running on the field" or "the football team are running on the field." Here whether you consider the team a group or a bunch of individuals tends to be different for different dialects and contexts. However, there are no dialects or context where "were" is appropriate for "period."
BTW, with regards to "data/dataum" and similarly "die/dice" this is a change in language over time. Datum and die have become rarely used and are dying out, so that the putatively plural forms have become uncountable nouns in common parlance. Not everyone will agree with this assessment, it is contextually dependent, and in some circumstances the difference is still adhered to. However, I think many people would agree that that is the direction the language is going with these words, notwithstanding the argument as to how far they have gone or how fast they are going.
Wiktionary on data and Wiktionary on dice list both as uncountable as one meaning, though the plural meaning is also indicated.