A number of falafel/gyro stands are open 24 hours, in particular the 3 Testver Torok Etkezde chain (3 Brothers' Turkish Fast Food). 

There is a 24-hour McDonald's at Oktogon square.

Both Nagyi Palacsintozoja locations are open non-stop.

Any market marked "non-stop" or "ejjel-nappal" is a 24 hour convenience store where you can pick up snacks and drinks.