This series on asexual forms of intimacy is part of my activity for Asexual Awareness Week.
Sleeping & Other Spacial Closeness
Everyone has a bubble of space around them, its size varying by culture, but usually about arm-length. Strangers and casual acquaintances customarily stay outside that area. Friends and coworkers will touch or slightly overlap edges. Only close friends, lovers, and family members tend to come into very close physical proximity. This is especially true in terms of sleeping, sitting, or traveling in the same space. In fact "sleeping together" is a euphemism for sexual intercourse, precisely because of its intimacy. However, that intimacy can be just as deep -- or deeper -- without involving anything sexual at all.
Putting someone to bed. Interestingly, this activity can happen among people who are just getting to know each other -- most often if someone passes out drunk, but exhaustion can have a similar effect. It's a gesture of caring to put someone to bed rather than leave them where they drop. A milder version involves draping a blanket or coat over a person asleep on a couch or the like.
Sleeping in the same bed. This is an act of shared vulnerability and intimacy. Lovers customarily do this; so do some siblings or friends, especially as children. People may also be driven to share a bed, sleeping bag, etc. for warmth or lack of other accommodations in challenging circumstances.
Watching someone sleep. There is more vulnerability on the part of the sleeper, and more intimacy from the watcher, when only one person is asleep. Parents often watch their children sleep. Lovers sometimes do this with each other, which can be cute or creepy. It's also a guard position, useful for showing that one character seeks to protect another.
Waking someone up from a nightmare. A subtler form of rescue than more physical actions, this is still a gesture of protection and caring. It often leads to comfort afterwards. A typical courtesy between parent and child, or lovers, this can also be an early threshold for characters thrust together unexpectedly if one of them has sturdy daytime walls and a lot of issues. It is common, but often unspoken, among war buddies or veterans, many of whom have nightmares.
Camping or hiking overnight. You wind up sharing a tent, if you're lucky enough to have one, perhaps a blanket or a pile of leaves if you're unlucky. Long-distance wilderneering pushes people to rely on each other as well as share space and more intimate awareness.
Sharing a saddle. Riding a horse or other animal requires a careful coordination of two bodies; adding a third makes it even more complex. The motion usually causes two people to rub against each other constantly, and fighting it throws everyone off-balance. Either you learn to cooperate very closely, or you wind up very uncomfortable. Friends often ride together; lovers and family members sometimes do; but this can also happen with strangers meeting during a rescue. It's a good way to push standoffish characters together.
Sharing car/berth space on a long trip. This is less intimate than riding, but still involves relatively close contact over an extended time. That usually gets people talking, a terrific icebreaker early in a relationship. In established relationships it offers a chance to spend time together and catch up on news.