I have stores hay in my barn for the last 15 years without a problem. There are two concerns you can take care of by following some basic practices when putting up and storing hay. Not only should you be conerned about the fire risk but there is a health concern for your horses from hay that is not processed and stored correctly. Hay that is baled before it is properly dry in the field and not picked up quickly enough after it has been baled is the main problem for both barn and horse safety.
Too much moisture in the hay is the key issue for both fires and mold growth in hay. Horses can develop deathly health problems from moldly hay. Moisture is what causes hay to get hot to the point of combustion in the barn. All hay will go through a heat cycle after it is baled, the amount of mositure in the hay determines how hot the hay will get. Of course warm damp conditions are the ideal environment for mold growth also.
Hay storage is a factor also. The ideal place will be dry, well vented, and no direct sunlight. That is one of the reasons hay was stored in the upper portions of barns. The hay was up off the ground, hay on the ground can pick up moisture from the ground. The siding on old barns was not all that tight allowing the barn to breathe. The upper area of a barn did not recieve direct sunlight which can bleach out the nourishment in the hay.
When you purchase hay stick your hand down in between the flakes you should be able to feel if there is too much moisture in it. Pick up the strings of the bale, wet hay is heavy and the string tension will give you an idea how heavy the bales are. Unless the bales are not packed that tight, you will kow how tight they are when you try to put your hand down into the bale. Look at the field, is the ground damp or wet. Did the hay come from the outer edges of the field where maybe tress shaded the outside egde. Around here, New Hampshire most fields a surrounded by tress, the outside windrow is usually the last baled because it dries the slowest.
If you put up your own hay, or have someone else bale it for you make sure the hay is properly dry before you bale it. Grab a hand full and squeeze it into a ball it, should not even think or staying in a ball. make sure the hay is baled well after the morning dew has dried off of it, and baled and picked up before the evening dew has started to settle.
There are as many ways to pile hay as there are people who store it. My favorite is to pile the hay bales on edge cut side up. Make sure you have good ventilation, air flow through the hay storage area, any hay is bound to have some mositure in it that will dry out over the winter when the humidity level is generally lower. Placing hay under tarps may increase the chance that the remaining mositure in the hay or from the ground will condence on the cool inside of the tarp and the hay will start to mold.
The requirements for dairy animal hay is different than horse hay, so pay particular attention to hay processed for dairy animals. When I put hay in the barn and come to a bale I think is questionable it gets put a side, only the best gets into the loft. The first few days I will go and put my arm in between the piled bales to see if I can detect any warm areas. These are just some of my experiences.
Hope this helps,
Randy