Although a vaccine for hepatitis C virus (HCV) remains elusive, new direct-acting antiviral (DAA) drugs can cure HCV effectively and conveniently. The latest DAA, Epclusa, can cure 95-99% of chronic hepatitis C regardless of genotype (e.g. see here). So if DAA drugs become universally available, will HCV be the first virus eradicated solely by drugs?
Is it feasible to eradicate a virus without vaccines?
It is possible? I would say yes. But feasible... I think not. Consider this, once vaccinated, a person is usually immune to the virus. And you only need the vaccine to take hold, for an individual to acquire life time immunity. However with drug treatment... even once the treatment succeeds, the person can be reinfected. And you will have come back with drug treatment again.
Given the difficult in vaccination campaigns (ie polio) to eradicate the last reserved of a virus... I think a similar drug based campaign would a lot more difficult. There is no herd immunity to prevent the virus from reinfecting the wider population when it leaves its strong holds.
To the question I would say certainly yes, since the only viruses that have been eradicated (to my knowledge) are smallpox and rinderpest, both via vaccine campaigns (as well as the usual tools of quarantine/isolation).
Note: I did not mean to suggest that "if drugs are widely available they will definitely eradicate HCV". I interpreted the question as "if HCV is eradicated by DAA, will this be the first disease eradicated by drugs?"
Vaccines are typically used as prophylactic treatments, i.e. they are applied before an individual gets infected. Drugs can also be applied prophylactically (e.g. antiretroviral drugs against HIV), and vaccines can also be used post-infection, e.g. for rabies. In order for a cure (i.e., a treatment that is applied after an individual becomes infected) to eradicate disease, it must be applied soon after enough infection that it cuts down the infection period sufficiently to reduce the average number of secondary infections per infected person below 1. Analogously, when prophylactic treatments are given by age, as in the case of vaccines for childhood diseases, they must be administered to children young enough that most of them will not already have been naturally infected [and passed the virus on to others].