Hgb 10.3 is serious problem for health?

Depends on cause: Hgb is the concentration of hemoglobin, the major oxygen carrying pigment in whole blood. 10.3 is lower than the average level 11.6 to 12. 3 for females. The next question is why is it low? Are you bleeding?Are the red blood being destroyed? Is the bone marrow suppressed? Are you iron deficient? Do you have a B12 or folate (folic acid) deficiency. Answer these questions. Treatable problem.

Anemia: You have anemia, though not severe it can become so. It is not unusual for menstruating females to get iron deficiency anemia. You need to see your doctor and make sure that is the reason for your anemia. If it is supplemental iron will fix this problem.

Find the cause: Being "a little bit anemic" is like being "a little bit pregnant". Your body's not going to break down just from this. But it's a warning that something's wrong. This may just be monthly iron loss or the effects of ill-advised vegetarianism. But it's a bleeding ulcer or stomach / colon cancer until proved otherwise. Please comply with the recommended workup.