Do DNA repressors exist?
Yes, these sequences exist and they are called "silencers" (surprising, right?). There are different mechanisms by which this silencing of genes can happen.

In the "classical" way the silencer is bound by a  transcription factor which either passively suppress the gene by hindering the binding of specific transcription factors or by actively preventing the assembly of the general transcription factors. See the figure from paper 1:



Additionally there are non-classical negative regulatory element (NRE), which are usually elements upstream of the promoter which inhibit the binding regulatory proteins. NRE can also be enhancers depending on the proteins bound on them. Some NRE can induce a bend of the DNA, inhibiting the access to enhancer or promoter elements.

References:


Transcriptional control and the role of silencers in transcriptional
regulation in eukaryotes.
Transcriptional Regulatory Elements in the Human Genome