So I remember there being a Marathon mode in Civ III, and at least one of the popular reviews talk about it. I can't how ever see any options for this? Any help is appreciated.

I haven't heard anything about a Marathon mode, and I've had the complete edition of the game for years.

Could they have been talking about the "campaign"? It is a series of maps going through different historical eras.

Maybe I'm wrong, maybe it wasn't in the third one?

"Marathon" is the slowest game speed in Civ IV. Everything takes more turns to complete (units, buildings, technologies, border expansions, tile improvements...) so the game lasts longer, you spend more time on each era and units can be used for a longer time before they become obsolete. There are four speeds (Quick, Normal, Epic, Marathon).

However, I have never played Civ III so I don't know if it has this feature.

Nope, Civ 3 has nothing like this. The closest thing would be the "accelerated production" option, which cuts the number of shields required to build anything in half.

There is an option to start a game in an advanced age, but I haven't ever tried it out, so I don't know exactly how it works.

Sounds like Civ 4 is the game with the marathon.

there is no marathon mode in Civ 3, this is Civ 4 feauture.

anyway, you can made it through the editor, i done it 2-3 years ago changing 'tech rate cost' in world map config, and only after ~240-300 turns most advanced civilizations become into Medieval era. But with changing 'tech rate cost' also you should change cost of every unit and building, something like x1.5-2 from base value. You should made it because new technologies discovers very slow, AI and you will end with everything builded in every city very fast. One thing, what i cant create like in Civ 4 marathon mode - to slow down cities grow.

While not technically a marathon mode, you can change the turn limit of the game (Game Limits options) to almost double the normal, so from 540 to 1000.
This will allow you to play the game for scoring purposes, double the amount.