Eukaryotic genomes are packaged into chromatin, which is composed of condensed filaments of regularly spaced nucleosomes, resembling beads on a string. The nucleosome contains ~147 bp of DNA wrapped almost twice around a central core histone octamer. The packaging of DNA into chromatin represents a challenge to transcription factors and other proteins requiring access to their binding sites. Consequently, control of DNA accessibility is thought to play a key role in gene regulation. Here we measure DNA accessibility genome wide in living budding yeast cells by inducible expression of DNA methyltransferases. We find that the genome is globally accessible in living cells, unlike in isolated nuclei, where DNA accessibility is severely restricted. Gene bodies are methylated at only slightly slower rates than promoters, indicating that yeast chromatin is highly dynamic in vivo. In contrast, silenced loci and centromeres are strongly protected. Global shifts in nucleosome positions occur in cells as they are depleted of ATP-dependent chromatin remodelers, suggesting that nucleosome dynamics result from competition among these enzymes. We conclude that chromatin is in a state of continuous flux in living cells, but static in nuclei, suggesting that DNA packaging in yeast is not generally repressive.