It would depend on the underlying cause of the low Hemoglobin. I have seen Hemoglobin in the 3's from a slow intestinal bleed, many times from colon cancers. The person was given transfusions, the location of the bleed was found so it could be stopped.
I have seen Hemoglobin in the 3-4's of women in their 50's and 60's who didn't have enough iron and vitamins in their diet, and who sought treatment merely because they were tired. They presented with pale hands and lips, but otherwise were not having noticeable symptoms. So in those particular cases nothing "happens" but they are given transfusion(s) and nutrition consults.
In cancer patients who are undergoing treatment, Hemoglobin dips low due to aggressive therapies to kill the cancer, such as chemotherapy and radiation. The type of cancer, as well as the physician, (or sometimes guidelines of a protocol), decides on the threshold at which to transfuse. Pediatrics as well as young people can tolerate a lower Hemoglobin than most.
Some oncological physicians choose not to transfuse until the patient is exhibiting symptoms, as the risk of transfusion may outweigh the benefits. Transfusions can cause iron to deposit in the heart, as well as the in the liver, which can cause cirrhosis. There are also infection risks in receiving a transfusion. Depending on the specifics involved, I have seen some physicians choose to wait to tranfuse a patient until they exhibit unfavorable symptoms, most often in a blood cancer. With cancers other than blood, the goal of the chemotherapy is to rid the body of cancer, and the low blood cells are an unfortunate side effect. With blood cancers, the low blood cells are the desired effect.
Also, Jehovah's Witness patients in treatment for cancer have religious customs which forbid blood transfusions. In those cases, I recall a patient who had a Hemoglobin of 2.8. That patient was in her 50's and she was instructed not to exert herself for risk of cardiac arrest. The patient didn't feel anything out of the ordinary with such a low Hemoglobin, and did not experience any resulting cardiac problems.
When answering this question, there are so many facets to be able to explain "what happens" is too specific to the individual person/situation.
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