Is storing 256GB using printed coloured geometric shapes printed on paper possible?

We think not but we could be wrong. It certainly appears that current printing and scanning technology cannot cope. Interestingly Xerox has worked on a denser form of barcode called a DataGlyph. These are a form of extended barcode with alternating small diagonal lines (pictured).

Xerox DataGlyph

Forward slashes represent a binary one and rearwards slashes a binary zero. To encode the data in a robust manner, the data is first converted using a Reed-Solomon code with two CCIR check bits. It is then interlaced to deal with burst errors, resulting from damage such as a paper fold or a mark on a glyph) and finally it is enmeshed in a synchronization frame.

They can store hundreds of times more information than ordinary barcodes, according to Xerox. But they can't store information needed for a 256GB/A4 sheet density. Without details of the encoding method used to produce the Rainbow format we don't know if the supposed technology is real, a mistake, or a piece of foolishness.