Depends on what you mean by "sequence a genome". There's nothing out there that can take a single chromosome and sit and spit out its sequence from one end to the other (even from organisms like yeast that have much smaller chromosomes). Instead, with high throughput sequencing you sequence between 30 and 150 base pairs at a time, and then stitch several million of those reads together. This computationally difficult problem can be made easier if you know the approximate spacing between two given reads, so you can connect islands together. Illumina (and possibly SOLID, but I'm not sure) lets you do paired-end sequencing, where you get both ends of a particular DNA molecule, and depending on how you set up your samples, you can have mate pairs separated by 3,000-5,000 base pairs. In order to get a really good de novo assembly of a genome, you typically want on the order of 50-100x average coverage of the genome. A single sequencing lane (which costs ~$1300) can get you about 5x coverage of a genome.

It's often possible to tell how many times a gene is repeated because, statistically, you should get fairly uniform coverage of the genome (although there are lots of artifacts in the sequencing process that makes this less true). If you get significantly more reads from a particular locus, it's likely that it's been duplicated.

Glancing at the figures in the article, more than 65% of this "structural variation" seems to be insertions or deletions of less than 10 bp, with only 2% coming from elements greater than 1,000bp.