Yes, it is possible to put up your own hay without a lot of equipment. Break it down into activities, cutting, curing, windrowing, gathering and storing.
While small plots could be but by hand, basically all you need is a bushhog. Just have it run low and slow. It will cut the hay up finer than a haybine, but the blades will do basically the same job of cracking the stems for drying.
For windrowing, as mentioned above there are ground-driven rakes. Basically two types. On one you have to raise the drawbar to travel or make corners. Second you have to raise or blower the reel. Mine is the latter, but I still use a crossbar on the three-point hitch to use it. (There are probably PTO driven ones as well, but I am not familiar with them.)
For curing you can let it lay as cut (the bushhog will scatter it some), then windrow it twice, with drying time inbetween. If you cut after the forages have dried one morning, you should be able to rake up windows in the early afternoon the next day and the second time towards the evening. Load up the next day. While three days of good sunny weather is great, this process can be crammed into two. Just don't put up more than you can handle at once.
For gathering, if you do the windrows right, a pickup and/or trailer can travel between two of them with one on each side pitchforking into it. If you put up temporary sideracks on a pickup, you can get a good bit of hay into it. Here a younger child can be packing as it is toss into the bed.
For storage, under cover, put down a layer of pallets and then loose stack the hay on them, putting down a fairly even layer each time. Don't walk on the hay while stacking. It would be better to have it settle on its own. Just put on a couple of loads at a time.
Ken S. in WC TN