Are all the biochemicals that our body uses enantiomerically pure or are racemic mixtures too?
Many molecules exist in both forms in nature. One fun example are the enantiomeric terpenoids R-(–)-carvone and S-(+)-carvone. The R-form smells like spearmint while the S-form smells like caraway. The difference in smell shows that properties other than the optical activity are different for two enantiomers.
Why, then, is one enantiomer preferred over the other?
The_Vinz stated correctly in the comments that for replicating structures like RNA, choice of the enatiomer was random; once established, one chiral form prevailed. DNA building blocks have the same chirality as those of RNA because they are made by the same biochemical pathway.
One amino acid, glycine, is not chiral. Many amino acids are found in both forms. L-amino acids are used to make proteins, but D-amino acids are made in bacteria and used in the context of cell walls and natural antibiotics.
Proteins made by ribosomes (from amino acids attached to tRNA by tRNA-synthetases) use L-amino acids exclusively. The tRNA-synthetases are highly specific (including stereo specific), and they don't link tRNA to D-amino acids. It helps that there are very little D-amino acids made in a typical cell. Why one form was chosen over the other is probably luck again. Why all amino acids have the same chirality at the alpha carbon is more intriguing. Some are made from the same precursor, so that will contribute. Right-handed alpha helices require that the amino acids in them be L-amino acids. If proteins had a mixture of L- and D-amino acids (e.g. all alanines are D-alanines but all aspartates are L-aspartate), alpha helices would be more constrained in the possible sequences, and some of them would be left-handed.
All other molecules
Most steps in the synthesis of biomolecules are catalyzed by enzymes. Enzymes, as chiral catalysts that have lots of interactions with reactants, are often highly stereospecific. So the presence or absence of enzymes catalyzing certain reactions largely determines which products are made, and there is no additional cost of making a chiral product from non-chiral precursors (very different from a typical lab synthesis).