Is this a joke?
That said, it's not my field and I get the impression they're talking more about in vitro work on the page I linked. Even if your circuits work all the time in the lab, in a cell I'd expect all kinds of things to mess with them.
In the former, you design sequences of DNA such that complementary base pairing means they can displace each other in cleaver ways. This lets you create some interesting things, like oscillators [1], amongst others [2]. These do not need any of the apparatus of the cell to function, so work in solution; indeed, if they were inside cells they would get digested by nucleases. The thermodynamics of DNA/RNA folding is fairly well understood, and the range of structures in much more limited than that of proteins. A major drawback of these circuits is that they function very slowly.
By 'genetic circuits', people usually mean a genetic regulatory network [3] - essentially you combine existing genes in new ways, by chaing the regulatory sequences before each gene. For example, you can construct an oscillator from three genes by having the first repress the second, which represses the third, which represses the first [4]. Here you aren't designing new proteins (which is extremely hard), but rather modifying existing ones. Since these circuits require producing new proteins from DNA, they require RNA polymerase, the proteosome, ATP, the necessary monomers etc. so can only function inside a cell (or cell-free expression system containing these components).
[1]: https://www.researchgate.net/publication/50304896_Programmin...
[2]: http://research.microsoft.com/en-us/projects/dna/
The problem isn't so much in designing the circuit abstractly as finding specific parts with which to construct it. One approach is to partition the circuit across multiple cells [0, 1].
[0]: http://www.nature.com/nature/journal/v469/n7329/full/nature0...
[1]: http://journals.plos.org/ploscompbiol/article?id=10.1371/jou...
In prokaryotic systems there is (in a very approximate, generic sense) a one-to-one correspondence between the concentration of a particular transcription factor and the expression (or repression) of the genes downstream of the binding site for that transcription factor.
The control regions in eukaryotic genomes have binding sites for multiple transcription factors, combinations of which may become binding sites for other transcription factors (larger TFs which bind to certain combinations of smaller TFs), etc.
In this way, the specific sequence of TF binding domains in the regulatory region of a eukaryotic gene provides a particular and potentially unique "address" in "Transcription Factor State Space" by which the gene can be controlled.
For more information on this amazing topic, check out "The Regulatory Genome" by Eric H. Davidson. Here is an excerpt from the first page of chapter 4:
"Whatever their extent, however, development gene regulatory networks have an internal structure, in that they are composed of diverse kinds of modular parts and connections among these parts. Here 'modular' takes on a simple functional meaning: it is used to denote small subsets of genes within the overall network that together execute given 'jobs,' e.g., to operate a certain differentiation gene battery, or to transduce an extracellular signal into a certain regulatory state.
In what follows, sets of regulatory genes that execute modular functions are usually referred to as constituting 'subcircuits' of the network, because as we shall shortly see they are 'wired together' within the subcircuit by their gene regulatory interactions. Just as the target site inputs of an individual cis-regulatory [note: cis- prefix in this context indicates gene regulation via non-expressed sequences of DNA proximal to a gene in the genome] module are integrated to generate novel outputs according to its genomic design, so the outputs of these subcircuits are integrated to generate logic outputs which depend on their organization, that is, their wiring architecture."
- https://books.google.com/books?id=F2ibJj1LHGEC&pg=PA126
[edit 1: added link to google books & excerpt]
Surely, also one has to account for possible other "jobs" that get done due to interference.
However, if your goal is to automate processes rather than develop cures to "run" in the human body then this is a very interesting alternative to using silicon, the parallel pipeline potential is enormous.
EDIT: Would it be possible to develop a biological CPU this way? I.e. having "instruction sensors" and a touring-machine-like DNA-robot that can execute externally supplied instructions? Putting that into a bacteria that can clone itself would surely cut down on costs of computing.
No, it is not possible (not this way). Tl;Dr how do you plan on storing information on the Turing machine tape? If you're happy doing computation with a relatively high stochastic failure rate things look better, but I wouldn't count on it.