How big of a bankroll do I need to play $1/3 hold’em?

It depends on your expected win rate, how much variance you expect to encounter, and how much risk of ruin you want to accept. If you have a large 10bb/hr win rate at live $1/$3 cash games, you can get by with only $7,000 or so. As your win rate decreases, you need much more, though.