RNA is known to act as an enzyme via its ability to fold itself in specific ways.
Is DNA capable of such structures? Or is it some biochemical reason stopping the folding? Have they been observed in nature?
There are no known natural DNA enzymes (deoxyribozymes), but there are various synthetic DNA enzymes. The first deoxyribozyme that has been reported (Breaker and Joyce, 1994) catalyzed the Pb2+-dependant cleavage of RNA.
Various deoxyribozymes have been synthesized, they can catalyze RNA cleavage, RNA ligation and many other reactions like DNA phosphorylation or Thymine dimer photoreversion, and even a Diels-Alder reaction (see Baum and Silverman, 2008 for a review).
DNA lacks the 2'-hydroxyl group that RNA posesses, but there is some evidence that this doesn't significantly decrease the potential of deoxyribozymes compared to ribozymes. In one experiment DNA and RNA enzymes that catalyze a Carbon-Carbon bond formation were compared and both achieved comparable catalytic rates (Chandra and Silverman, 2008).
For more information about DNA enzymes you can look at the publications from the Silverman lab, they're probably the most active research group in this field.
DNA has recently been designed to act as a targeted drug-delivery agent at George Church's lab. This doesn't involve catalyzing a reaction between two substrates, but it does involve significant internal conformational changes based on binding to highly specific substrates.