I know that "long-shot" roughly means "improbable", but I am not sure what a long-shot job is, at least in the context of searching for such a job. Is it a really odd job, a job where the appliant has very few possibilities of being hired, or something else?
A "long-shot" in terms of a job search is similar to any other "long-shot" attempt at something. It is a job that the speaker could apply to, but has little chance or expectation of getting. As an example, consider a generally unremarkable fresh graduate from college saying the following:
[Major multinational company] just fired their CEO and is looking for a new one. It's a long shot but I think I'll apply!
In this case the speaker is probably being humorous, but you could use this term to describe applying for a position for which you are somewhat, but not completely qualified, in the hopes of impressing the hiring manager and/or recruiter enough to get the job anyway.
Point of clarification: as @Chokkeliet mentioned, "long-shot job" sounds odd and is not necessarily a phrase that is often used, however you can describe a job as being a "long-shot"
Hope that helps
long shot (plural: long shots)
(idiomatic, nautical) Something unlikely; something that has little chance of happening or working. The term arose from the accuracy of early ship guns, which were effective only at close range and unlikely to hit the mark at any great distance. We can try your plan, but it's a long shot and it probably won't work.
Though, in general, I'd consider "long-shot job" to have at least two clearly different meanings (as opposed to a job that is a long shot). The first is it means that it's a job (read: occupation) that's really hard or difficult to get. The second is it means that it's a job concerned with working on things that are hard or difficult to complete or finish.
Unless, however, a "long-shot job" is some kind of slang that I've never heard before...
A "long shot" is a gambling term, typically used in referring to horse races.
In the U.S., wagering on horse races uses a "parimutuel" system for determining the value of a winning bet. Increased wagerng on a particular horse will cause the amounts paid to winning bettors to decrease; diminished wagering on a particular horse will cause the amounts paid to winning bettors to increase. The more a horse becomes unpopular among bettors, it becomes more of a long shot, and the odds paid on a winning ticket might increase to 20-1, or 30-1, in other words winning ticket holders might be paid $42 or $62 on a $2 wager.
Usually (but not always) the actual probability that a horse will win the race correlates with the wagering odds, due to the "wisdom of crowds." So the term "long shot" refers to an improbable event; and also to the fact that, if wagering is permitted on the outcome of the event, an increased payment is had if one bets on the improbable event and it occurs.