Can I share my contacts list with family members?

Not with Apple, but you can set up a separate Gmail account for this e.g. <family.surname>@gmail.com.

There you add your mutual contacts, or export / import them from your computer.

Then add this gmail account to your device [Settings => Mail, Contacts Calendars] and switch on the contacts button, to make them visible in Contacts as well.

Then in [Settings => Mail, Contacts Calendars] scroll down to the contacts section and choose as Default Account for contacts that Gmail Account. That way, all new contacts will by default be created in this 'family' Gmail Account.

Your other family members should do the same.

There. You have your shared contacts account.

I'll add that if you don't want to share your entire contacts list and just some select contacts, you can always use AirDrop (provided that your devices support it) to share individual contacts.

See the following for more info on AirDrop:

Share content with AirDrop from your iPhone, iPad, or iPod touch - Apple Support