MCCOY: I thought [Spock] was the only one who's immortal.
KIRK: Oh, no, it isn't that. I knew I wouldn't die because the two of you were with me.
Although this sounds a bit emo, it's Kirk's way of expressing how much he relies on his crew—and Spock and McCoy, in particular. These guys have been together for decades, after all. As we'll see, their experiences together have transformed them into a little quasi-family.
MCCOY: All that time in space and getting on each other's nerves, and what do we do when shore leave comes along? We spend it together.
Please note that during this scene, Spock, Kirk, and McCoy are on a camping trip together. As Bones observes, this is legit insane. These three men have spent practically every waking moment together for decades. Why would they spend their free time together, too?
MCCOY: Other people have families.
KIRK: Other people, Bones. Not us.
As Starfleet officers, the crew members of Enterprise have little chance to have personal lives. They have trouble enough staying in touch with the family they have, much less starting ones of their own.
[Spock is unable to make himself shoot Sybok, despite having him at gunpoint. Sybok takes the gun from his hands.]
Spoiler: Sybok is Spock's half-brother. Brain blown. This scene is important because it shows that although Spock is loyal to Enterprise, he still cares for his brother enough to spare the dude's life—even if that means losing the ship in the process.
[McCoy sees his father suffering from a grave illness and agrees to perform assisted suicide to release him from his pain.]
Our brief glimpse of McCoy's relationship with his father is devastating. Through Sybok's magic-therapy powers, we learn that McCoy unplugged his dad from life support when he was deathly ill, in an attempt to ease his pain. Heavy. Even now, McCoy is wracked with guilt and unsure if he did the right thing.
[In a hallucination, Sarek sees his newborn son Spock for the first time.]
SAREK: So human.
Here's some context: Spock's dad, Sarek, is a Vulcan, while his mother is human. In this light, it's clear that Sarek is throwing shade at his newborn son and distancing himself from him.
SYBOK: Spock, Dr. McCoy, come with me.
[Spock doesn't move.]
SPOCK: I cannot go with you.
Despite his telepathic therapy sesh with Sybok, Spock still refuses to join his bro's team. For whatever reason, he sees his bond with Kirk and the crew of Enterprise as more important than his bond with Sybok. Why do you think that is?
SPOCK: Sybok, you are my brother, but you do not know me. I am not the outcast boy you left behind those many years ago.
Sometimes, when we look at family members, we can get caught up thinking about them as they used to be, rather than as they are now. That's a big mistake—and more than a little lame. It also happens to be exactly what Sybok is doing with Spock.
SPOCK: I was thinking of Sybok. I have lost a brother.
KIRK: Yes. I lost a brother once. I was lucky. I got him back.
This, if you don't know, is a reference to Spock's death and subsequent resurrection in Star Trek II and Star Trek III. The exchange emphasizes the familial nature of the crew's relationships with each other.
MCCOY: I thought you said men like us don't have families.
KIRK: I was wrong.
Bingo. Throughout the movie, the crew members of Enterprise prove their loyalty and love for each other on countless occasions—and how else do you define "family" except through loyalty and love?