Bedrock is breakable in survival, but you can't get it as an item. The way to break it is more of a glitch in the programming, because technically it's not supposed to be breakable. It doesn't require many materials to break it.

Materials you will need to break bedrock:
4 Dark oak saplings
15 Bone meal (amount varies but 15 is a good amount to bring to be sure you have enough)
4 blocks of dirt

What you should bring on the journey and to do the process:
The materials listed above (You will need those)
6 pickaxes (Digging down and clearing an area)
4 shovels (Digging down and clearing an area)
Flint and steel (For burning the tree)
axe (For breaking the tree)
Sword and armor (In case you encounter monsters)
Food (don't want to starve while down there)

How to do it-

Step 1
Get to bedrock
Bring a few pickaxes and shovels and dig down to the bedrock layer with the materials for breaking bedrock in your inventory.

Step 2
Clear an area
You'll have to clear an 8x8 square area by destroying all the the blocks around it. You will also have to dig out the blocks on the roof of the 8x8 square about 10 blocks high (If it doesn't work with 10, dig a bit higher). (This is why you need multiple pickaxes and shovels.)

Step 3
Break the bedrock
In the center 4x4 area in the 8x8 space you cleared, place the 4 dirt blocks. On top of those 4 dirt blocks, place all 4 saplings. *Make sure the saplings are dark oak. Then use bone meal on one of the sapling until it grows into a tree.

Step 4
Find the broken bedrock
All you have to do is dig through the bottom of the tree and below the 4x4 are where you placed the dirt, 1 or 2 pieces of bedrock will be broken. (If for some reason none are broken, one thing you can do is light the tree on fire until it burns away and you can try again. Or you can just bring an axe and chop it away.)

That is the ancient way to break bedrock.