Disclaimer, I'm no pro in fermenting, I just do it exactly like my wife's grandpa did - and it works. If you are a smartass and know how to do better, then please post a guide on how to improve fermenting and production.
To make a kvass mash:
Take a loaf of rye bread (any non purely wheat bread should work, but rye bread works best - I use whole grain rye bread), slice it in even, 1cm thick slices.
Bake them in the oven until they are brownish color (you could likely skip the baking and sun-dry it, or something. The point is to dry the bread up reeeal good). It stores well.
Take filtered water in a big 5l jar, and add sugar (5 tablespoons should be enough, but you could use more) and some yeast (really not much, it will eat sugar and grow anyway). You will only need to add yeast once.
Put the bread into the jar as well. Cover with gauze, leave for 3 days.
After 3 days, drain it (it will be rather disgusting to drink, but will smell good - don't drink it though), take bread mash, divide in half (you can either start a new batch or just throw that half away) and add half of the freshly dried loaf.
Add sugar. Again, don't add anymore yeast. There is no need.
Wait for 3 more days, drain again. You now have a good partly fermented mash that will give you good kvass from now on.
Cycle is like that -
1. Take mash that fermented at least 2 times, divide in half, one half goes into trash (or use it with other batch)
2. Add half of baked/dried bread loaf.
3. Take clean water, 5l
4. Add sugar, at least 1 tablespoon per liter of water, and stir well until sugar is dissolved.
5. Put mash and bread into sugar water, cover with gauze, wait 2-3 days, drain, enjoy (after first day you can taste it - drain when the sweetness is appropriate, store in fridge, and drink in the next 24-36 hours - after that it's just not very tasty anymore).
6. After draining, remove mash, goto 1.
Mash is stored in the fridge - I never had problems with using weeks old mash - no weird growths or anything. It just eats all remaining sugar and goes into anabiosis.
You may try using natural sugars (like apple juice), it may or may not work good. I will try some day when I will have enough space.