Hay and straw are two different products. Hay is the long grass cut which includes the nourishing seeds and is used to feed livestock. Straw is the stalks, usually of grain crops, after harvest, which makes good ground cover for livestock in stalls and barns. Once swept up after livestock have relieved themselves on it, it can be added to composting remains.
Since you only have a small suburban yard, perhaps share it with a gardening neighbor. An interesting fact is that people use straw, NOT hay, bales if the old rectangular type for building sheds, homes and emergency shelters. Google “straw bale home or building construction” to see some very interesting building uses for those old style rectangular bales. There are some 100 year old homes that have survived.
Straw bale gardening is catching on...we are going to give it a try this year after Helene displaced us into a more ‘suburban’ setting (spit!).