Are Aloe Plants Related To Cactus Plants?

By plant standards, fairly distantly related.

Cacti are any of the plants in a number of genuses in the family of plants called Cactaceae.  Cacti are green succulents with thick skins and a large water storage capacity inside; they can also survive drying out fairly well.  They usually have spiny protruberances to safeguard their water supply water from thirsty animals.  Cactaceae is in the taxonomic order Caryophyllales, which also includes carnations, spinach, rhubarb, bougainvilla, buckwheat, amaranth, Jojoba, tumbleweeds and many others.

So Cactus and Jojoba, another favourite plant from which to extract skin ointments, aren't too distantly related.

Aloes include over 400 species, of which Aloe vera is the most commonly known.  They do have thick skins, short spines and high water storage capacity, but this a genus within the family of plants called Asphodelaceae.  Most of the plants in Asphodelaceae are flowery types.  Asphodelaceae is in the Asparagales order of plants, which also includes garlic, onion, hyacinths, irises, orchids and a variety of ornamental flowering shurbs.