http://www.findlaw.com.au/articles/432/age-of-consent-to-medical-treatment.aspx
18 it seems (16 in SA, in special circumstances 14 in NSW). It varies between the states, and there seems to be some felxibility depending on the age of the individual, and perhaps with respect to the medical response being perscribed (ie. a doc might not worry if they were prescibing eyedrops to a 16 year old…but perhaps the legal risks there might stop them anyways).
But, regardless, seeing a doctor (regardless of whether they can medically intervene or not) sounds like a good idea regardless. He or she might help identify the cause of the problem (ie lack of sleep can cause depression), which might involve ceasing substances (ie. caffein, etc..), or changing lifestyle (eg. exercise, eating habits, reducing stress) rather than having a script filled.
Just be aware that they are likely to be legally obliged to involve your parents if active medical intervention is to be started, depending on where you are in Oz.