Even though I'm an atheist, ever since reading "The Name of the Rose" I've leaned towards finding monasteries cool for a variety of reasons.

First of all: monasteries begat universities. I think it can be asserted as almost self-evident that universities are among the most important and wholesome institutions that humanity has invented.

Second: monasteries held huge libraries. In these libraries some of the classics survived, and most of the original research of the time was developed or copied.

Third: monasteries were ran according to a fixed scheduled adjusted to lithurgy. I don't usually like fixed schedules, finding them draconian; but for some reason I am drawn towards this one:

https://en.wikipedia.org/wiki/Canonical_hours

Naming hours and assigning them a function seems useful.

Perhaps following a fixed schedule with distinctive associated rituals can promote something close to mindfulness.

Fourth: monasteries were often self-contained and self-sustaining. Nothing drives this point home like a visit to Meteora, which I've had the fortune to see:

https://en.wikipedia.org/wiki/Meteora 

Using Hans Widmer's terminology, monasteries offered an alternate "deal" to those lucky enough that might also be interested in partaking. You could opt out of mainstream feudal society and buy into a life of structured work and devotion -- or study.

In exchange for relative independence, monasteries did not usually meddle with the affairs of the people. In some times and places, they might as well have been hermetic spaceships.

Many were, of course, hierarchical places full of fanatism and cruelty and oscurantism. But, if you were an intellectual in the middle ages, you could certainly have done a lot worse. They were at least a place where you could find like-minded people.

An Agora.