It can work, but be very, very careful about ventilation and air circulation. It's not (well, very seldom) an issue for feedstores and large barns that get frequent new deliveries, but if you are going to put months' worth of hay or feed in there, you can run into problems in the later parts of winter and early spring.
What happens is, the hay etc in the trailer (or shipping container) gets very very cold over the middle of the winter, as does the container itself, and then you get a February thaw type day or some balmy March breezes and the warm(er) humid air hits the cold air mass inside the trailer/container and BAM it's a condensation farm.
This is especially a problem in trailers/containers IME because they are usually hard to ventilate well, and lack much of anything that could absorb the moisture (as dirt, gravel, cement and wood can in a normal barn). So once the condensation is there, it is THERE, and can quickly mold up all the rest of your hay, and yes mold does grow in the cold.
I'm not saying don't do it -- it's probably better overall than storing hay under a tarp, if there is no indoor option -- just saying to be very aware that this will happen, and try to arrange the stacking and useage of the hay to maximize airflow, especially against the walls. And to keep an especially close eye out for problems, come March or so.
Good luck, have fun,
Pat