Storage is not the problem. Computational power is.
Each genetic sequence is ~3GB but since sequences between individuals are very similar it is possible to compress them by only recording the differences from a reference sequences making each genome ~20 MB. This means you could store a sequences for everybody in the world in ~132 PB or 0.05% or total worldwide data storage (295 exabytes)
Now the real challenge is more in having enough computational power to read and process the 3 billion letters genetic sequence and designing effective algorithms to process this data.
More info on compression of genomic sequences