Moonbear said:
Antibodies don't ... target DNA, they target proteins.
What Moonbear said is mostly accurate, but some antibodies can indeed target nucleic acids. For example, an (autoimmune) antibody to double-stranded DNA is a common finding in systemic lupus erythematosus (SLE).
Antibodies can target pretty much any molecule of sufficient size/complexity. But (as Moonbear meant) antibodies produced by the body to target pathogens (like viruses and bacteria) are mainly directed against the epitopes (you should look this term up) on the exterior of the pathogen.
Antibodies come in various sizes dependent on their class. The "usual" antibody type people visualise when they talk about them is the IgG class (G-class immunoglobulin) which has 2 heavy chains and 2 light chains bonded together. IgM (M-class immunoglobulins) are the biggest and heaviest in their native state because they occur in a pentameric form with 5 units, each like an IgG molecule, giving a total of 10 heavy and 10 light chains.
You can make fragments of antibodies that retain the binding specificity but are lighter and smaller. These are called F_{ab} fragments and they are sometimes used to bind and mop-up excess drug molecules to treat overdoses (like in digoxin toxicity).
VIral structure is more complex than you seem to think.

They may have a lipid envelope that sheaths the nucleocapsid (in which case, the capsomeres or capsid proteins won't be exposed on the surface). Embedded in the lipid envelope may be lipoproteins or glycoprotein "spikes" which also can elicit an antibody response for e.g. when you talk about the flu (influenza), antibodies form against hemagglutinin and neuraminidase molecules which are prominent projections from the viral surface. Non-enveloped viruses will have their capsids on their outside but may also have these spikes on their surface, e.g. the glycoproteins poking out from the vertices of the icosahedral capsid of the adenovirus.
Another thing you must keep in mind is that not all antibodies are "neutralising" (blocking the virus' attachment and propagation through steric hindrance) - some work in other ways (like complement activation or "opsonisation" - making the virus particles "tasty" for macrophages which gobble them up). Still others serve no good function, they are simply ineffective products of the immune response, and some of these may actually be harmful to the body because of immunological cross reactions with healthy body tissue (the basis for one form of autoimmune disease).
Antibody mediated immunity is important to many viruses, but in some viruses it just doesn't work enough to prevent a reinfection. With nearly all viruses, some degree of "cell mediated immunity" is important. This has nothing to do with antibodies per se, but a lot to do with T-lymphocytes. This is a big topic, I suggest you do a search on this as well.
I hope I've answered some of your questions (even ones you haven't asked yet), and pointed you in some new directions. Immunology and Microbiology are fascinating subjects and the latter happens to be my bread and butter.
