We know that reprogrammable self-replicating systems are possible because they're swarming all around us. Every living thing is a reprogrammable self-replicating system. DNA is the program, asexual or sexual reproduction is the means of replication. But there's still more work to be done before we can create artificial self-replicating systems. Let's take a look at the history of the concept in recent times.
The concept of self-replicating automata was first formalized by John von Neumann, one of the greatest computer scientists and mathematicians of the 20th century. His Universal Constructor was virtual rather than physical, and can be seen as history's first computer virus. Von Neumann proved that the most effective way of performing massive mining operations such as mining an entire moon or asteroid belt would be by using self-replicating machines, taking advantage of their exponential growth. His magnum opus on the topic, Theory of Self-Reproducing Automata, was published in 1966.
After von Neumann's work, the field of self-replicating systems was dormant for a decade and a half. It was revived in 1980 at the request of newly elected President Jimmy Carter, for a cost of $11.7 million. This was the landmark "Advanced Automation for Space Missions" NASA summer study, conducted by Robert Freitas, among others. The study, which focused on lunar robotics, concluded, "there are several alternative strategies by which machine self-replication can be carried out in a practical engineering setting", and that "the virtually cost-free expansion of mining, processing, and manufacturing capacity, once an initial investment is made in an autonomous self-replicating system, makes possible the commercial utilization of the abundant energy and mineral resources of the Moon". Unfortunately this proposal was quietly declined and passed into obscurity, with negligible media coverage. Freitas still works on the tools to build artificial self-replicating systems today, through his Nanofactory Collaboration project.
In the early and mid-80s, an MIT graduate student named Eric Drexler made waves with his theories of nanoscale assemblers and self-replicating nanobots. His landmark 1986 book, Engines of Creation, has since been translated into six different languages and serves as a standard reference for nanotechnology discussions. In 1992, he authored a more technical book, Nanosystems, which goes into great detail regarding the feasibility of self-replicating molecular assemblers. In the 15 years since this book has been published, its critics have yet to find a single technical error. This was the first book to show that ribosomes are not the only theoretically possible molecular-scale self-replicating assemblers. It also showed how massively parallel manufacturing by molecular assemblers could be used to build human-scale products to atomic precision, without overheating, in durations measured in hours. The main challenge would be building the first molecular assembler - from then on, the exponential power of self-replication would take over.
Most recently, in 2004, Tihamer Toth-Fejel determined that "the complexity of a useful kinematic self-replicating system is less than that of a Pentium IV". This conclusion came out of a NASA Institute for Advanced Concepts study, "Modeling Kinematic Cellular Automata". In 2003, for their Timeline for Molecular Manufacturing, the Center for Responsible Nanotechnology (CRN) argued that the complexity of a self-replicating molecular assembler would be similar to that of the Space Shuttle. In his October 2003 paper, "Design of a Primitive Nanofactory", CRN Director of Research Chris Phoenix explained the development path between the first molecular assembler and desktop nanofactories in great detail, showing how tiny 200 nm blocks could be combined in enormous numbers to create human-sized products made out of diamond, using nothing but simple hydrocarbons for feedstock.
Today, the prospect of reprogrammable self-replicating machines is all but ignored by mainstream science and engineering. A dozen or so researchers continue to pursue the goal, hampered by a lack of funding and popular support. The community is small enough that I've met most of the involved individuals personally and continue to correspond with many of them regularly. Most of these researchers believe we'll be able to build a molecular assembler sometime before 2025.
For much more on this, see my other posts on the topic. For a comprehensive view on the history and theory of self-replicating kinematic machines, see the book of the same name, Kinematic Self-Replicating Machines. Incidentally, First Class members of the Lifeboat Foundation get this book sent to them free of charge.