Do It Yourself

I'll preface with my qualifications, so if a more qualified person comes along you can disregard me. I'm an engineer who has taken a few thermodynamics courses and has worked as an engineer for a hvac manufacturing plant. I've never done anything strictly related to geothermal, but I've read a decent bit about it (and watched Technology Connections' video on the subject, it's a good entry point)

You may want to call up a company who does geothermal cooling and see what options you have, they've gotten pretty creative on how to bury the cooling lines. (See the video mentioned before)

Going the route of just sticking a large water tank underground probably won't do a ton. I expect that you will have a poor surface to volume ratio, which means poor heat transfer, which means you'll saturate your thermal mass fairly quickly. What this may allow you to do is run your HVAC system during the night/morning when it's much more efficient, and 'charge' your thermal mass for the hottest part of the day.

Assuming you use 300kg of water in a day, and you can get a 10°C delta, my very rough back of the napkin math says you're only going to have about 3 kWh of cooling from just the cold water, which is a decent bit, but it's not a ton. Best case scenario you cut your cooling needs by around 10-20%.

I'm too lazy to do the math of the heat exhange with the ground, but my bet would be you're better off spending any money you have set aside for this on better insulation techniques and/or a proper geothermal cooling system.

I do like your creative idea though