It sounds like you're mainly using hay as animal bedding, and possibly for a bit of food for the rabbits? With respect to mulch in the summer, it would make sense to put the used bedding to service as mulch -- it will carry along a bit of extra nutrient enrichment. (Of course, being careful not to use it around crops that would come into contact with manures like spinach or lettuce.)
My choices for bedding would include straw, wood shavings, sawdust, hay, and dry leaves -- depending on the needs of the animals.
With any of these, if you are going to DIY, you can make a place to store the finished product until it is ready for use.
Starting with the cheapest and possibly the easiest: dry leaves. You might be able to acquire a large quantity of leaves for free. Do you live in a place where people rake up the leaves on their lawns and leave them in bags at the curb? If so, round up a bunch of bags when they're in season and store them for use as bedding.
Depending on where you live, you might be able to find wood byproducts from a mill for cheap or free -- you'd just have to find a way to transport them to your house. Unfortunately arborist wood chips, which can sometimes be had for free by asking a passing tree truck, are too coarse for animal bedding -- though they do make for nice mulch.
You could try growing your own straw. I believe you're in Australia's hardiness zone 4, so your winter lows are about 0°C? You could set aside a portion of your yard to growing a grain crop for straw. An appropriate choice might even be something you could grow as a cover crop during your mild winter. With some planning and luck maybe you could even get a triple yield and harvest some grain for use as animal feed or for your own use. (But at least save enough grain so that you have seed for the next crop!) The downside here is the amount of space required, the low straw yield you'll probably get, and the amount of manual labor required to process it. At your scale a scythe is a reasonable way to harvest the crop; I'm guessing that machinery isn't practical in your setting (e.g. hiring someone to come and process 700m2 seems very unlikely). The straw also won't be compacted like machine made bales, so it will take up more space in storage, and may be more prone to taking up moisture if you live in a humid climate.
Similarly you could try planting your yard to a hay crop. This has the benefit over straw of being perennial. It may be harder to process -- grass is wetter and will take longer to dry. It is also heavier to handle. I'm not a fan of hay for mulch since it is full of seeds and often carries weeds.
If you don't want to do it yourself, but want to avoid going to a store, ask around to find farmers in your area who may be willing to sell you what you need. You may be able to get it cheaper or more conveniently if you have the storage at your house to buy a bulk amount.
You can usually save money if you pick up the hay "out of the field" instead of after the farmer has done the work of transporting it and stacking it in his barn. This takes some coordination with the farmer so that you know exactly when the hay is being harvested, and being flexible with your schedule to go and pick it up immediately. (I do this, and it sometimes requires clearing my schedule for an afternoon so that I can go pick up hay and stack it in my barn.)
On a final note: you will probably end up wanting to combine some of these. E.g. build a storage shed, get dry leaves off your neighbors' curbs when you can, plant a stand of buff oats for straw and as a cover crop, and find someone nearby who can supply you with enough hay to fill those months where you have gaps in your homegrown supply.