In terms of pure economics, it might be possible. Philosophically, communism is materialistic and anti-theistic, but if we were to trash all of that (as we rightfully should) and only consider the economic theories of marxism, it might be possible.
If you think about it, the traditional Christian family unit (as described in The Bible itself) is communistic. We have a father, a mother, and children. The father is the provider who brings home the bread and distributes the resources among his family, the mother who does not work still earns her stay by doing house work and caring for the children. The children cannot earn their stay, but receive everything they need anyways. This looks to me like Karl Marx's own rule, "From each according to their ability, to each according to their need" being followed perfectly. There are ascetic monks who live in completely autonomous monasteries that practice a sort of communism in the sense that everyone does work, and everyone receives the same amount of resources. These monasteries are usually very primitivistic and agrarian, the introduction to technology into this system, I think poses many problems, and also there is the problem with volunteerism. The monks do this voluntarily because they knew the lifestyle they were choosing when they became monks. If one of these monasteries were to be converted into a commune where people will raise their children, there is no guarantee that the newer generations will follow this practice and take advantage of the system.
In conclusion, I think it is possible for Christians to follow marxist economics (but definitely not marxist philosophy or social policy) without violating Christian ethics, and a Christian nation could potentially institute a marxist economic system, but that system would fail so it's largely pointless to think about.