Can I make the saving throw from Relentless Rage (PHB p.49, Barbarian) and use my Relentless Endurance (PHB p.41, Half-Orc) if I fail it? Is there anything that would suggest this is not possible?
Yes, you can use both features.
The key is that, in both features, the text states that "if you drop to 0 hitpoints and don't die outright, you can use this feature..."
So, when you drop to 0 hp, you choose to use relentless rage, and fail the saving throw. Now, you're still dropping to 0 hp, so the conditions are right for you to choose to use relentless endurance.
Supplement: I was not able to find any errata pertaining to these features. With no additional information from the developers, I see no reason to assume anything other than the clear reading of the text as we have it.