It is not safe to store hay outdoors.
Not in the Fukushima area, anyway. Japan and its farmers are finding this out the hard way after radioactive beef was found in stores in Tokyo. The beef had 5 times the legal limit of radioactive caesium. This is not an alarming level, but it is also not a pattern you would want to continue.
Food authorities traced the radioactivity back to grasses that were stored outdoors and took on radioactive dust from the air. A ban on Fukushima beef will likely be announced in a few days after the details are worked out.
If the grass is radioactive, it seems fair to guess that plants of all kinds are radioactive. This is a problem that can last for decades. Germany is still finding radioactive wild boars on a daily basis. The boars become radioactive after eating mushrooms, which concentrate the radioactive minerals released in the Chernobyl disaster. It may take another century in Germany before anyone would consider hunting boars without a Geiger counter. No one can yet say when the Fukushima area will again be a place to raise beef cattle.