Here are some saves.
This is just my 2nd civ 3 game so don't laugh too hard.
Advice #0: In the game preferences make sure that "Always wait at the end of the turn" is checked. Otherwise playing is a little pointless, as the game will just end the turn regardless of whether you are finished or not. It is a bit unfortunate that the default setting is to "Off".
My lands are devoid of luxuries.
I have 2 fur and 1 gems. That's it.
Not quite. There are Gems. But the English have them. But you could easily steal them, by placing a city in this way:
It is a little close to P'yongyang, but I would not worry about that too much. OCP is not very effective. Since it will also be fairly close to your capital you should see that you turn it into a core city with Library, Aqueduct, University and so on.
Then, I find that there is no iron.
Arabia declares war because I wouldn't give away free items they requested.
They send Medieval Infantry, so they have iron.
They even have an iron
to spare. So, if you were at peace and had a trade route you could import one of their iron sources. However, at the moment you don't need it that desperately anyway, really.
I figured I was dead but my spears and archers did well holding them off from inside the city.
Then I got long bows that I sent to the city along with Trebuches which did well by bombarding them.
Yes, you are doing quite OK against them. It is not really such a big problem holding back the enemy with Trebuchets and Longbowmen.
I raced to gunpowder only to find that there is no saltpeter anywhere.
How can you say there is no saltpeter if you know hardly anything about the map?

In addition, ask yourself, is saltpeter really that important? Unless you want to conquer the world with Cavalry, it is pretty negligible. At the moment you are not really in a shape to do much conquering anyway and Military Tradition is something that you will get to in the mid run. Apart from Cavs, Saltpeter allows Muskets which you don't really need, Frigates and Privateers which you don't really need either and which are a thing of the future. Hwach'as ... ok, that is your UU, but they also have only limited use too, especially since you have already had your Golden Age. (Right?? You have the GLib + Collossus and you are Korea.)
Iron is slightly different. True, you don't really need it at the moment that badly. Arabia is not that much of a problem: you could easily continue to trash their stream/trickle of MI that are coming your way, or you could sign peace as they are willing to talk and accept straight up peace. However in the mid run, you will want to be able to build railroads, and for that you need iron and coal. Arabia has iron to spare, and that is at least an option to keep an eye on.
How am I supposed to play?
In the situation that you are in, I would say continue the game with a higher emphasis on the builder angle. Scientific progress is your best bet. As Korea, Libraries and Universities are cheap and will help you a lot. Build more cities in the land that you have and improve the land as quickly and well as you can. And explore the world. You need to make contacts so you can trade for and sell technologies, and so you can import luxuries and resources.
That means, your empire needs:
-- Many more workers. You can hardly have enough of them
-- Settlers. You can easily build twice as many cities as you have now and then a few.
-- Boats to explore. Later maybe a few explorers which you can use to investigate the territory of your neighbors (England etc.).
-- Libraries, (Aqueducts,) Universities, Markets, (Harbours,) and Courts. Roughly in this order of importance. Granaries can be good here and there for cities tasked with building workers and settlers, preferably at size below 7. The rest is not so important.
-- Trade routes with your neighbors and civs you hopefully are going to meet.
-- Forbidden Palace.
Let me have a look at your cities:
Seoul
You are building a University, which is exactly the right decision. Afterwards, think about building a Harbour. This would give you more food from the coast tiles, and allow you to mine some of the irrigated grassland. Apart from that you don't need anything else improvement wise, at least not now. Since you have the Collossus here,
and it is on the verge of giving you tourism bonus shortly, you could really think about turning this city into a Super Science City. Means, build Copernicus Observatory and Newton's University.
If you opt for an SSC, then Seoul will fully be occupied with building those two wonders, and you can sell the granary and the barracks.
P'yongyang
This city is not growing and stuck at size 9. It needs to have some of its plains irrigated. It also needs a Library and a University. As of now you are not nearly tapping the research potential of that city. It can give you three times the number of beakers than you have now.
It is relatvily commerce rich and can also be really productive and eventually produce the bulk of your military.
Wonsan
That is the city which is under attack. However, Arabia is not a big problem anyway. The city also has a happiness problem and is in disorder.
I would suggest using that city to produce settlers and workers. It will shrink, but that is OK. Switch the Longbow to a Settler. And, just for this one turn only, turn several of the laborers to scientists:
That way you'll get a nice, one time amount of beakers. If I am not mistaken, the setter will complete next turn. After that, continue building settlers and workers. Wonsan has already got a Granary and is quite good foodwise.
Pusan
Pusan is building a Harbour. That is a good decision. After that, a galley or two to explore. And then a University. The Harbour will help the city to grow more quickly, and you should let it. Eventually you can place some more mines.
Namp'o
This city is building the Sistine Chapel. I am not sure you really need that. It is not such a good wonder. The Forbidden Palace would do much more for your Empire. You'll lose some shields if you switch, but that is OK. Afterwards, one Settler -- the city can regrow easily. And then a university.
The city has plenty of food but needs more shields.
Cheju
The city is producing Wealth. Not such a good decision. Maybe spit out a worker, then a galley, and afterwards build it up fully, with Library, University, Harbour. Marketplace and Courthouse are already present.
The city is relatively commerce rich, what with the three sugars, the river and and coastal tiles.
Hyangsan
This city is building a Cathedral. If you switch to an Aqueduct that will complete immediately. Afterwards, use the city for a limited time to build settlers and workes. And then build it up with the necessary improvements.
Pyongsong
Dunno. Build workers maybe.
Inch'on
Settlers and workers.
Taejon
A fledling Core City. It is building a Lib and that is a good start.
So I guess I'm going to quit this one.
Is this a common occurrence to get so totally screwed on resources?
Definitely no need to abandon the game. In fact, I think it is easy to win.
Luxuries and Resources can sometimes be quite tricky to obtain, but the resources are sometimes not really needed, and there is often the option to trade. No need to hesitate. And who knows, maybe there is an island or so that has what you need.