Can I replace the entire hose bib stem?
The way you get a new stem is to get a whole new sillcock/hose bibb. 

If the stem is that corroded, the body probably is as well, so it's likely to not seal right even if you could fine one to match a specific 34 year old valve body (which seems like a bit of a stretch.)

If you can identify your particular valve you MIGHT find a specialist "repair parts" plumbing supplier that has parts, such as (maybe) this one; http://www.azpartsmaster.com/faucet-valve-stem.aspx but if you cannot positively identify the valve, that's unlikely to work (and if the body itself is bad, it may not work anyway, as already mentioned.) If you need a place you can walk into with a valve stem you may need to take a LONG drive to find one "near" you that goes beyond just washers.

At some point fairly early in the process (IMHO) replacing the whole valve makes more sense.