It all depends on what you mean by "own".
The major issue is that of patents of individual genes, particularly in the United States. A study in 2005 published in Science suggested 20% of human gene DNA sequences were patented and that some genes were patented as many as 20 times. But (in theory at least) each patent is not of the gene itself, but of its use for a particular purpose. A recent example in the news was the Myriad Genetics patents on the BRCA1 and BRCA2 genes, linked to diagnosing breast cancer, which were upheld by the US Federal Circuit Court of Appeals.
But this is not ownership in the standard sense. The company cannot sue you if you reproduce your genes in the natural way. Nor can you sue the company if one of their genes in your body damages you.
There may also be copyright issues if you have your full DNA sequenced and that too has been discussed. But it is much less likely to be a significant issue, as you can always have it resequenced, rather in the way a photograph of a mountain may be copyrighted but this does not prevent a different photograph of the same mountain.