Antibodies will stay in your blood forever. It is the same process as getting measles or chickenpox and then recovering-your body produces antibodies against these viruses. Antibodies kill the viruses-which means recovery for you. Consider yourself very lucky. In most people who get hepatitis C can not fight it off, and the disease becomes chronic.
Antibodies can not cause any harm to you. There are hundreds of antibodies in your blood-produced as a result of disease or vaccination.
I don't think that presence of antibodies can disqualify you from service in the military. You don't have hepatitis C-that is what really matters. But knowing how poorly informed health care professionals are about this disease, if they try to disqualify you, you might need to educate them and to provide some documentation which shows that you have just antibodies, not the virus.
Yes, you will always have the antibodies in your body. If they have any concern about it all they have to do is a test for live virus, and that won't be there if you have truly fought it off.