Can you be goth and emo?

No.

A Goth is someone who's predominantly a fan of goth rock (Joy Division, Bauhaus, Sisters of Mercy, The Cure, Siouxsie and the Banshees etc)

and an Emo is someone who's predominantly a fan of Emotive Rock (American Football, Weezer etc).

A misconception about both subcultures is that they're based on a mindset or attitude (most people think emo stands for Emotional, when it doesn't. Most people think goths have to be dark and introspective).

You can't predominantly be a fan of two separate and unrelated Music Genres at the same time, although you can be an Emo who likes some Goth Rock, and a Goth who likes some Emotive rock.