You have a toilet that needs minor repair, say replacement of a float or flapper. For purposes of this question, assume that whoever lives there owns the problem-- there's no super, maintenance guy, etc.
It's really easy to flush a broken toilet - all you need is a bucket of water. Slowly pour the bucket into the bowl and it will flush. (The flushing mechanism operates based on the water level in the bowl.) All the tank mechanism does is "pour" the water into the bowl for you, and refill the bucket (tank) automatically.
Keep this tip in mind if you are ever without running water for an extended period of time.
I was without running water for 17 days over Christmas and New Years after a part in our pump froze and broke in the big freeze a few days before 12/25/91. (The company that made parts for our pump was closed over the holidays and didn't reopen until after 1/2 which delayed our part order, and then we were on a long list for the plumber to come out and install the part.) Fortunately, our pump had been drawing water out of the top valve in our 1000 gallon tank (located mid-way down the side of the tank) so we still had ~500 gallons in the tank. We put a plastic trash barrel on the front porch and my Ex bailed water from the tank to fill the barrel. Then we bailed from the barrel as needed to bring water indoors for flushing the toilet and to boil and then use to wash dishes. We had a neighbor with city water (we were on our own well) who let us use their guest bathroom for showering and use their washer for our laundry. It was a bit of an ordeal, but not all that difficult, to live without running water for 17 days. It was definitely an adventure!
The hardest part was the first few days when I didn't have water for the horses. They were on the city water system, which took 3-4 days to patch and restore water to all addresses. We had to put barrels in the back of the pickup and drive to the one working public tap in town, fill the barrels, and then drive to the horses and transfer water from the barrels to the horse's water tubs.