Yes, we are at an early stage with close systems like that. First stage surely is to get them working on the Earth. Then in space, in LEO or somewhere accessible so that we can get back in an emergency and fix them with resupply from Earth.
We've made some progress on the Earth. With experiments going back to the 1970s
This is the NASA biohome. Not an entirely closed system, not closed for air, but showed the possibility of plants to clean the air and for food, and of bulrushes and so forth to clean wastes.
The Russians did a more closed system project, Bios 1-3, where they generated all the oxygen from algae - it needed 8 square meters of algae per person. Biospheres. The facility still exists. BIOS-3 (Wikipedia)
which didn't quite succeed in creating a closed ecosystem - and is more of the scale of a really large space settlement - but mainly due to issues with the concrete removing oxygen through chemical reactions. Plus management and personality issues. And it was way more complex than an early closes system space station ecosystem would be. Biosphere 2
So - basically I think we are at very early days. The experiments so far don't seem to have turned up any essential obstacles to make it impossible. Closed ecosystems are certainly possible.
But is it possible to have a closed ecosystem that has humans inside it? Or an almost closed ecosystem? And with everything recycled including human waste?
All the indications seem to be that it is possible, and the Bios-3 get close to it, and has had people live in it continuously for several months, depending on algae for oxygen. But none of them inhabited continuously for years on end, at least not successfully. And none of them have been flown in space, which has its own unique challenges.
So - I think all the deep space missions that rely on closed system ecosystems as if it was a an already mature and developed technology - there is a gap there, something that needs to be proved first. And the bugs worked out (literally also, because one of the problems with biosphere 2 was single species such as ants taking over parts of the ecology). If you depend on a closed system like that, you are also vulnerable to diseases of the plants and pests and such like.
In space you have the extra issue also of construction, their greenhouses have to be built to withstand about a ton per square meter of outward pressure. And would operate at a lower pressure, of a tenth of Earth normal atmosphere, in some of the plans (if at full pressure, they have to withstand ten tons per square meter of outward pressure). So you'd need to test your closed ecology ecosystem under those conditions also of tenth atmospheric pressure, if that's the plan.
The CO2 in the Mars atmosphere doesn't seem a great advantage to me for a closed system ecology, as it is normally a waste gas you get rid of in a space habitat, the ISS continually vents CO2 into space.
You'll see from my other answers, I think that in any case we won't be able to colonize Mars in near future for planetary protection reasons. That Mars is a connected system because of the global dust storms of dust particles that block UV light and can transfer spores from anywhere on Mars to anywhere else. So how can you introduce microbes to part of Mars and not infect the whole planet? And the main interest in exploring Mars is the search for life. In the ideas with human astronauts on the surface, humans go nowhere near the areas which may have life, to avoid contaminating them. So they send clean robots to those areas and operate them remotely, which bring back samples to the base. So - why not have the humans in orbit instead, operating the robots telerobotically from orbit, since the whole thing has to be done remotely? I don't see the benefit in humans being on the surface, myself, and impossible to keep a human Mars base clean of Earth microbial life, obviously, so huge planetary protection issues.
And in my view, there is no value in Mars as a "second home" for Earth because you just need to look at the solar system, there is no way you'd choose Mars over Earth even if the very worst happened to the Earth, no disaster could make it as uninhabitable as Mars. You'd want to come back to Earth and restore it, it is the only thing that makes sense for future of humanity, not stay on Mars and try to keep going there when the Earth is there, even in devastated state, if that was what happened. So it's no insurance in my view at all. Not until 500 million years from now when Earth may become uninhabitable naturally - and who knows who will be here and what will be needed by then (long enough for humans to evolve a second time from the very first multi-cellular creatures) - and Mars if made partly habitable now, would have long lost its atmosphere and volatiles by then.
Far better to care for our Earth. And best place to put insurance colonies to survive disasters, is again on the Earth, right where you want them to be to help restore the planet - deep below the oceans or on remote islands or whatever - if you think they are needed at all.
But Mars is a great place to explore and to study, and the safest way is to do that from orbit, from Deimos, or from an eccentric orbit that dips in close to Mars twice a day, spectacular orbit, and operate clean robots on the surface via telepresence. I think that some of the Mars One applicants might well find that as attractive a mission as the original plan to land on the surface. So I'm hopeful myself that perhaps, if the Mars One or Space-X plans continue, maybe they may morph into a telepresence / orbital mission? Which also makes return to Earth far easier and the whole thing safer for the astronauts.
But first we need to sort out closed system habitats - or some way of at least reducing dependence.
If we start to get e.g. Bigelow habitats or whatever in LEO that are able to sustain themselves with almost closed systems, then deep space missions using that technology will seem more immediately possible :).
But as it is now, with only the ISS as a model, which doesn't attempt to be closed system, we don't have the technology, not in a mature state, I'd say.
If they did attempt a closed system ecosystem like Bios 3, it would be interesting and could potentially help with developing the technology in space. But I expect they will just have a base with humans wearing spacesuits though not needing them, in facilities that are just ordinary buildings, so not testing the technology needed to keep alive in a space station, just the psychological issues of living in those conditions. The Mars Society often does simulated missions like that, so it is a natural and easy thing for them to do.
Which are interesting for other reasons, but not really doing much for developing the technology of closed ecosystem type systems.