You will need 27 stacks of 64 cobblestone which you get from mining stone and then after placing it all into a chest at once.
Using a few iron or a diamond pickaxe would be best for this. Especially if you have efficiency enchantments on your pickaxe.
For those who don't want to count out 27 stacks just drop 3 full rows of 64 into a single chest at once and poof!
If you already have mined 27 stacks or more of cobblestone and it's already in a chest just take one stack out at and place it all back into the chest and the achievement should pop. If you haven't previously mined the 1728 cobblestone you must mine before the achievement pops.