Is having a constant pulse rate of around 120bpm for the last few months bad?

It's not normal: When resting, your heart rate should be less than 100. Metabolic conditions can lead to inappropriately high heart rates and you should be evaluated for these by your doctor.

Not normal: That would definitely be very elevated, although it's hard to know if your hr is really that high persistently without use of a monitor (e.g., a holter study). The most important issue would be identifying the reason for the elevated heart rate and addressing that problem, although at some point, medical management to lower the rate might be considered.