How big could this cave room be, given these constraints?
Romans seem to have dugs mines in the size of 2.2 km long before stopping and 200 meters in depth, often just stopping at the water table. There seems to be no reason to suspect they couldn't build a big dome shaped cave underground of any size given their other tech achievements and the scale we see here.
The size of rocks often shut down a mine, but that's because they were hoisting things up a deep shaft, they probably thought the mine wasn't worth any more time, etc, because they certainly could have broken a rock down if they really wanted to.
Besides this, to get a dome without a hole at the top you'll be using either a straight horizontal shaft or a vertical shaft down then horizontal shaft in. Either way this really isn't a limit to people who really want to do this, but to make it more time plausible let's say we go the horizontal route and that means that any cave they make can only be as big as the mountain they can get to, no bigger. That's massive, not helpful, and you're limited to where you can put it, and it's easier to spot with lower technology.
Let's say you wan to build underground rather than undermountain... Well basically, because we know that the water table and shaft depth is around 200 meters the answer to that is a ~200-400 meter diameter dome, based on their biggest dome they ever made (The Pantheon) which is 43.3 meters in diameters and height, meaning dome walls are very thick, but because we're already starting with rock material they should be able to make it any size based on the same principles as long as the material is dense enough...
So the answer in just size is...
- Limited to what we've seen: 43.3m
- Extrapolated from what we've seen underground: 200-400m
- Extrapolated from what we've seen undermountain: as big as the mountain.
Are the resources (manpower, time),
Here's the thing. We know that Roman soldiers could build cities very rapidly and certainly there was enough soldiers/etc. The issue is that they never undertook these projects, they were funded privately and really slowly with not the greatest labor. So when they built the Pantheon they likely didn't have a lot of labor nor quality labor nor was it a thing to get done right away... plus there is a difference between building a dome and cutting out stones to form one.
The Pantheon took some time less than 110 years to build. The inception of building was in the 30s BCE and it was partially destroyed in the 80s CE. so it had to have been finished some time in that period.
Assuming I'm right about how slow they were going and how fast they could go, my bet is it wouldn't take them that long to actually build any size. More over, we're dealing with just the removing part of the stone. We're dealing with roughly double the size of the pyramids with about the same amount of work, so I would say 20 to 40 years would be quite plausible for the underground 200-400 meter diameter dome.
If you build this farther out it limits manpower, closer in more man power. And you can just use the Chinese and Egyptian way of getting workers... it's a tax right off.
The biggest issues that you're going to face are...
- light (not realy since you can use candles)
- getting man power in the cave at first
- getting rock out.
Once it is built you're going to have to deal with as time moves on things get built and a cave under a city will eventually get dug into if noone knows its there. If you build it further out you solve that problem for a while but then lose man power.
Without more in depth knowledge of their mining abilities and architectural abilities, that's the extent, I think, of what a layman can say about this question...
Source for Roman mining info : Here