How to properly store cubed carrots?
The cubed carrots should be placed like the image. It should not be kept in water as the flavour of the carrots will decrease with time. I have noticed that the long term exposure of carrots to water will soften them but with some noticeable change in the taste.

Further, to keep the carrots really fresh you can use the same technique, simply wet a piece of cloth (preferably cotton), squeeze to loose 70% of water in it and then cover your carrots. The freshness and the taste; everything will be maintained for sure.