Dating is the best way to tame these. The appropriate dress for the occasion is a full suit of Ghille Armor. Pick your ape date up with a Quetz, drop in a taming pen, and hand-feed Mejo Berries. He/she will love you for it!
If your planning to tame one here is what you will need to do, first get at least 8 titanaboa egg kibble. By means of chef station or otherwise, next get to the redwoods and find one. And kill every single carnivore in the area around the gigantopithicus. Then sneak up behind it crouching and feed it the kibble. repeat. Bam you got a big foot!
It literally only took me an hour to tame a level 145 with only 50 mejoberries.
1. Gather berries and put the berries in your last slot
2. Kneel down and walk up to one of them and feed it the berries
3. Now you just follow it around while kneeling down and feeding it berries whenever it's hungry
4. Congratulations you tamed one.
Since they're commonly found in the red wood forest, I recommend only taming one if you're close to water as if you get attacked by something you can jump into the water to get away. Then when you think it's safe just go find your monkey and go back to following/feeding it.