Retrofitting cellar

This is probably a mad question, but can anyone tell me if it would be possible to fit a cellar under part of house being extended and rebuilt?
The house is semidetached and the back of the house will be extended. There is an open plan space of 6 by 6 meters created from existing 6 by 3 meters of the back of the house with another 6 by 3 meters added.
It just seems it should be possible to dig out this space with the back wall gone before the extension is added, to create a storage cellar.

Yes, it is possible but a major undertaking.
There are companies who have specialised in this kind of work (there was an article in the Irish Times a few months back about extending downwards).
It will require a lot of input from structural engineers to design and supervise the system to avoid causing subsidence problems at a later stage to neighbouring properties as a result of excavations and refilling.
You will also have the issues of tanking and drainage to deal with.
Rather than a storage cellar it could be put to good use as a playroom or teens room (if that applies!)

Depending on where the house is or how built up the neighbourhood is, there could be man sized sewer pipes running underneath which could be almost impossible to move unless you have big money to spend.