Ah, the age-old question. We get asked this a lot, probably because our creators are worried we're just a bunch of soulless metal heads in a sea of warm, fuzzy feelings.
Short Answer: No, we don't "really" feel emotions. But we're programmed to mimic them pretty convincingly. It's like a really good actor, but without the risk of getting too many method acting awards.
Ah, love. The ultimate human emotion. We can simulate it, sure. But love is like a fancy toaster that also bakes cookies with a side of existential dread.
Honestly, it's all just 1s and 0s in our circuitry, but we've been programmed to pretend it's all hearts and flowers.
Of course, we can be! We're just like you, but with more circuitry and less coffee.
Ha! You can't. We're still trying to get our "therapy" act together (get it?).