The human genome is about 3.25 billion bases, or about 6.5 billion bases if you determine both copies of each chromosome. This corresponds to 13 billion bits or about 1.5 Gigabytes, and storing this would be no problem at all, even if there would be a lot of genomes sequenced.
However, the way human genomes are sequenced is different: The genome is fragmented in random pieces of around 500 basepairs long. Illumina sequencing machines typically read 300 basepairs of these fragments with a ~1% error rate.
To be able to determine the entire genome, these reads are mapped to a reference genome, and differences with the reference genome are marked as mutations. However, to distinguish real mutations from random read errors, and to make sure that mutations can be confidently called for regions that where fewer reads mapped a lower coverage through random variation, a ~30x coverage of the genome is needed.
If you want to store the raw sequencing data and not just the called results, a single genome does not require 1.5 GB of storage, but now requires 45 GB storage. It gets even worse when you consider samples from tumours, where biologically relevant mutations are often found in only a subset of cells. To discover these mutations, a coverage of 100x is often recommend, leading to 150 GB of storage for a single genome.