We had no water at our last farm, I hauled it all to the farm in a 350 gallon tank in the back of my truck.
I generally started my garden very early each year to get as much spring rain as possible with plants already rooted well and growing. I would haul 5 gallon buckets of water to some of the plants from a surface well that I had dug. I would generally just water the tomatoes with about 20 to 25 gallons a day.
The peas, corn, radishes, beans, potatoes and garlic generally went without any water unless they started to look a bit wilted. Every year I managed to get great crops of peas, radishes, garlic and quite often potatoes. I managed to get corn one time in 8 years and it was only a few under developed heads.
One dry year I decided to try and water the garden with water from the 350 gallon tank, after doing that for about a week, I realized that I would need around 350 gallons a day minimum to make much of a difference.
My gardens are generally quite large, I would work up 2,000+ square feet of garden each year and practice trying to get vegetable to grow without anything but the natural rainfall. I wasted a lot of time, but I also learned a lot.
If you keep the surface of the soil covered with old hay it will help the soil retain moisture much better.
I also found that if I covered mounds of old hay that set out all winter with about a foot of good soil, the subsurface hay would hold a great amount of water from the winter well into the summer.
I also found you preferably want to water at night rather than during the day.
Having plants closer together and grown in thick like you see in nature seems to help the plants to lose moisture less quickly, but also tends to make them fruit and go to seed sooner, bearing smaller than normal fruit.
Keeping the plants somewhat shaded from direct hot sun in the mid day seems to help with water as well.
I also found that if I used a post hole digger I could dig down about three feet to the water table in spring and put good soft soil in all the way to the spring water table that seemed to help a bit with water.
As for your idea of using a backpack sprayer to water, all of my sprayers hold about 2 gallons of water or so, you will be refilling water a LOT even on a reasonably small garden. If I have to water anything by hand it is with five gallons buckets, I always use two buckets that way I can keep the weight on either side of me equal and tape garden hose over the handles so that it is easier on the hands.
If you were going to use some kind of a pump, I would advise an RV water pump,12volt and the ones I used to feed our house at the last place were rated at about 3.5 gallons a minute. I just ran our water from the cistern into the house with RV pump run off of a 12 volt battery that was always hooked to a charger. This was our household water supply for nearly ten years. The nice thing was, that when the electric went out we still had water pressure from the cistern to the house for up to three days running off the battery.
I have four large ponds here on this farm and a good well, so water is no longer an issue for us now. I have a pond that is about 30 feet higher than everything around the house and one of the former owners of this place used to gravity irrigate the young trees in the orchard and his garden from that pond. He warned not to forget about turning it off, he forgot and let it run for several days a few times and nearly ran the pond dry a couple times. This pond is triangular shaped and around 65 feet on each side with a depth of about 14 feet at the center.
What are the dimensions and depth of your pond that you are watering out of approximately? I can calculate how many gallons of water it is holding for you, handy to know if you ever had to rely on it for water.