Under what circumstances is working in the chemistry lab hazardous to health in the long term?
May I start with a disclaimer saying that whatever long term studies can be done, this will always be a subjective question. All I can do here is offer a few comparisons and examples to show that you shouldn't (in my opinion) be worried.

There answer to the question "Am I more likely to come into contact with carcinogenic/hazardous substances than a regular person?" is quite likely yes. At undergraduate level you will be using a range of strong, harmful, poisonous etc. substances that you could not get outside of a lab environment or at school.

However, the answer to the question "Am I more likely to suffer poor health as a result of working in a lab?" is more difficult.

As you correctly state in your question, there are always many safety procedures in place when you undertake experiments which have been carefully thought, require the scientists to have some minimum level of training in the techniques (perhaps learning with less dangerous substances) out and always err on the side of safe rather than sorry. I myself have been working with a new nanotube based material recently and though unlikely to be carcinogenic, all experiments were to be carried out using gloves in a fume cupboard with extraction as the material hasn't been fully classified or tested yet (again: "safe rather than sorry").

Of course, there are unfortunate cases (for example Marie Curie), but in this day and age the risk assessments are so extensive and always erring on the side of caution that the chance of something going terribly wrong are no worse, I feel, than something catastrophic happening in many other walks of life (e.g. mountaineering accident, plane crash, shark attack).

One perhaps reassuring and rather amusing case is that you are far less likely to succumb to terminal cancer if you work in the nuclear industry by virtue of the extensive risk assessments, precautions and health checks. There is so much safety in place at a reactor, that exposure is minimal (indeed, the background radiation from radon in Cornwall is higher than that next to a nuclear power plant!!) and the industry provides so many health checks for workers that and cancer (radiation induced or natural) is often picked up at a very early, curable stage. Earlier than if you simply went to a doctor when you began to feel ill as a regular person.

Therefore depending on the policy of your employer, you could in fact be at a lower risk of dying from cancer, and I see no need for you to worry for your health in the lab!!

As mentioned when I started, this is a subjective issue, but I really see no need for worry.

Best,

Jack