What's a noun for the group of people who you're very close to, such as family, friends, relatives, and significant others?

You can call all of them as family. Because you can also call your friends that is very close to you with kinship terms like "brother","sister" etc. It is not a true kinship but it is called fictive kinship.

Also, kinfolk (and folks) is used in this sense:

The people you hold the closest in life, or are related to.

As a phrase, there is kith and kin:

friends and relatives; people known to someone.

Also, kith itself means everyone you know but usually used in the above phrase.

familiar friends, neighbors, or relatives

^{Sources: Urban Dictionary, The Free Dictionary, and Merriam-Webster}

loved ones

nearest and dearest

I think that another common expression to indicate the group of people with which you have your main contacts in life is your circle namely, family circle and circle of friends

A slang Australian expression is "My mob". But that consists of two words.