Reddit reviews On the Frontier of Science: An American Rhetoric of Exploration and Exploitation (Rhetoric & Public Affairs)

Costs of sequencing DNA have gone down dramatically over the past decade. Creating a catalog of genetic sequences is now more feasible, technically speaking, than ever before. However there is more to understanding an organism than just its genetic sequence. Epigenetics play an important piece in how the genetic sequence of an organism gets translated into function. It is a much more time-consuming and costly endeavor to profile the entire epigenetic landscape of an organism.

The other problem, which can't be solved with money, is the problem of accessing Earth's biodiversity. There are some governments that view cataloging DNA of their biodiverse environments (like the rainforests in Brazil) as bio-colonialism and are therefore opposed to it. The politics of this situation is explored in the book On The Frontier of Science.

So where does that leave feasibility? Technologically, its possible. Funding-wise, DNA sequencing is probably possible, but epigenetics complicates this. But due to global politics, its unlikely that a complete catalog could be achieved.