There are some gems sitting in the middle of a mountain range, beset on all sides by what must be sheer cliffs. Are they forever out of reach or is there some tactic to sneak workers in?
You're out of luck. There's no way to get past the mountain range with workers.
Actually, if legions upgraded to paratroopers, and if you use e.g. the balance mod, which connects resources if there's a fort, you could connect the resource. Alas, legions are not on the right upgrade path, so you're still screwed.
If another civilization has a border close to the mountain range it might be possible to teleport in by declaring war with your worker positioned close next to the mountain in soon to be enemy territory. When the worker is evicted I believe it will move to the closest valid tile.
Not very practical though.
I know that this post is outdated. However, there is another method that has not been described. If every tile in your territory, save for the inaccessible tile in question, is occupied by a non-combative unit which includes great persons, settlers and workers, and if you were to create a worker at the city closest to the tile, the worker would spawn at the tile! Because Civ 5 does not permit the stacking of units of the same type, your worker would spawn at the desired tile. Thus, you would need to ensure that every hexagon nearest the tile and around your closest city is occupied by non-combative units when you create the worker and that there is no way to manually move your newly created worker via road, rail or sea to an unoccupied tile. Note further that you would need to ensure that the distance to this tile must be less than the distance to another non-occupied tile or your worker will spawn at the closer, unoccupied tile.
I have actually tested this method for the sake of curiosity. The same holds true for other unit types and permits effects such as spawning giant death robots in far away lands.
I tested goofballs solution, as I found it a little bit obscure. I noticed it doesn't work at all - might have worked in a previous version.
The current behaviour is the following: The only thing that matters is, if there is a hex that a newly produced worker could be moved from the city. It there is, then the new worker will always spawn in the city hex. If there already is a non-combat unit in the city hex, then the player is prompted to move the unit, and the game does not progress until the player does so.
However, if there is no hex the worker could be moved into (your territories borders are irrelevant - it could be a hex outside your territory), only then is the worker spawned in the nearest hex possible - i.e. not in enemy territory or occupied by an enemy unit (or barbarian camp, etc.). (again: if the hex(es) are inside your territory or not, is irrelevant). Sadly, even then there are limits - at most 2 hexes away, and even then it won't spawn it behind a mountain range - instead a message "no room for unit" is displayed and the unit is disbanded!
So do not waste time trying to improve a resource behind mountains as in Goofballs solution, does not work anymore ;-)
This isn't a general solution, but if you have the Gods and Kings expansion, you can do this as Carthage. After the first great general, Carthaginian units (including workers) can cross mountain tiles.