Ok.... so...... I'm kinda freaking out a bit because I saw the most GORGEOUS Gilgamesh walking out in the city today! It was so so gorgeous! I didn't even know they still bred those!
Unfortunately, though, they're rather hard to find and high-maintenance. A lot of owners only manage to find them through specialized breeders.
If you really want to get one...
It's generally considered bad practice to keep a Gilgamesh alone. If you're getting one from a special breeder, any of them worth your salt should be able to provide you with an Enkidu to keep them company, which will much improve their temperament.
Though they are also rare, if you're in with other "vintage" Servant crowds, you may encounter an Ishtar. Be wary, as they tend to not get along well with Gils.
Because of the old breeding, depending on who you get it from, your Gil may or may not have various health problems - make sure to ask first and be prepared if he needs special medication or other accommodations. Most Gils are generally good at hiding this in public, though, especially since lots of owners these days use them as a Show Breed.
Make sure to have lots of sparkly things around for them as a form of enrichment.
