Quite a lot, if you encode information in DNA [1] and then soak the paper in it. Densities of 5.5 petabits per cubic millimeter of DNA have been experimentally achieved. [2]

A typical sheet of A4 printer paper is about 6237 cubic millimeters in exterior volume (i.e. including interstices between fibers within the sheet.) Say you could soak a sheet of paper in soluble DNA and dry it such that you ended up with 6000 mm^3 of DNA in and on it. That'd be roughly 4000 petabytes.

[1] - https://en.wikipedia.org/wiki/DNA_digital_data_storage [2] - https://www.ncbi.nlm.nih.gov/pubmed/22903519