What family on the Periodic Table of Elements has a full outer electron shell?
The noble gases or Group VIII A
This family of elements is also known as the inert gases. The elements in this family have a full outer orbital shell.
The noble gases (group 18)
The group (family) with elements cotaining full outer shells are the right-most group on the table: the noble gases: helium, neon, argon, krypton, xenon, and radon (element 118, oganesson, also belongs in this group, but most all of its chemical and physical properties are as of yet unknown. We're not even sure if it's a gas!)
These elements are all generally inert and unreactive, and do not typically form bonds, primarily due to their valence shells being fully occupied.
However, some of the heavier noble gases have been observed in chemical bonds, such as
With a full outer shell of electrons, noble gases (or inert gases) have very high ionization energies (requires a lot of energy to remove an electron) and very low electron affinities (does not readily accept an electron), which explains their general lack of reactivity.