I did a talk on this a few years back at TEDx Austin (shameless self promotion): http://www.youtube.com/watch?v=8C-8j4Zhxlc
I still deal with this on a daily basis and it's a real challenge. Next-generation sequencing instruments are amazing tools and are truly transforming biology. However, the basic science of genomics will always be data intensive. Sequencing depth (the amount of data that needs to be collected) is driven primarily by the fact that genomes are large (e. coli has around 5 M bases in it's genome, humans have around 3 billion) and biology is noisy. Genomes must be over-sampled to produce useful results. For example, detecting variants in a genome requires 15-30x coverage. For a human, this equates to 45-90 Gbases or raw sequence data, which is roughly 45-90 GB of stored data for a single experiment.
The two common solutions I've noticed mentioned often in this thread, compression and clouds, are promising, but not yet practical in all situations. Compression helps save on storage, but almost every tool works on ASCII data, so there's always a time penalty when accessing the data. The formats of record for genomic sequences are also all ASCII (fasta, and more recently fastq), so it will be a while, if ever, before binary formats become standard.
The grid/cloud is a promising future solution, but there are still some barriers. Moving a few hundred gigs of data to the cloud is non-trivial over most networks (yes, those lucky enough to have Internet2 connections can do it better, assuming the bio building has a line running to it) and, despite the marketing hype, Amazon does not like it when you send disks. It's also cheaper to host your own hardware if you're generating tens or hundreds of terabytes. 40 TB on Amazon costs roughly $80k a year whereas 40 TB on an HPC storage system is roughly $60k total (assuming you're buying 200+ TB, which is not uncommon). Even adding an admin and using 3 years' depreciation, it's cheaper to have your own storage. The compute needs are rather modest as most sequencing applications are I/O bound - a few high memory (64 GB) nodes are all that's usually needed.
Keep in mind, too, that we're asking biologists to do this. Many biologists got into biology because they didn't like math and computers. Prior to next-generation sequencing, most biological computation happened in calculators and lab notebooks.
Needless to say, this is a very fun time to be a computer scientist working in the field.
-Chris