The thermal mass of concrete is actually trivial, compared to the thermal mass of water. If you go with a cheaper material and build a larger tank, you'll get more benefit.

A pound of water will store one btu for each degree of temperature change.

So, if you have a system which can operate on water as cool as 120F, and you have a tank which can withstand 170F as a normal maximum temperature, you have 50 degrees of temperature change (delta-T). If you want to be able to last for 24 hours on that tank on the coldest day of the year, and your house draws 40,000 btu's per hour on that day, then you multiply 40,000 by 24 hours, to get 960,000 btus, then divide by the delta-T of 50 degrees, to get 19,200 pounds of water.

Divide that by 8.3 (pounds per gallon of water), and you get a total of about 2300 gallons. If you want to be able to re-heat that with a four-hour burn, you'd divide the total storage capacity (960,000 btus) by four hours, to get a firing rate of 240,000 btus per hour.

If you're willing to do two fires per day on that coldest day of the year, then you can halve those numbers. If you need to be able to last two days (eg, you might be away on a trip), you'll need to double them. So, how you're willing to operate the system has a huge impact on how large it has to be.