The crew of the USS Enterprise had accepted that they had an unusual captain. Not that any of them considered that a bad thing, quite the opposite in fact. However, it did mean that he was constantly surprising them with stories and skills that they hadn't known he possessed. Once they thought about it, the crew realized that they knew next to nothing about Kirk's past besides that fact that he was born in Riverside, Iowa and had a mother in Starfleet, as well as the fact that he ascended the ranks faster than anyone in history, and held the record for being the youngest captain on record.
That being the case, they were always a little surprised if not shocked at the new tidbits of information that their captain allowed them to receive, and each person stored the information away to remember for as long as possible. Over the years, they grew used to the myriad of surprises, and began to enjoy the reactions of strangers and newcomers to their unusual captain.
Each person on board knew that their captain was unique. They knew that Kirk considered them family, and they felt honored at the title. They also knew that their captain would do anything in his power to keep them safe, and even a few things that logically shouldn't be possible. The crew members were what kept the Enterprise going, but Kirk was the glue that kept them together, and they all knew that.
Kirk's loyalty wasn't one-sided though. Every person would follow him to the edge of the universe and beyond, because they trusted him completely. That's not saying that his antics didn't irritate them to no end at times, but they knew that he would always put their best interests first without hesitation, a quality that was to be respected and treated like the gift it was. Indeed, there was not a member aboard that would ever regret the fact that their captain was James Tiberius Kirk.