There is a modest facility (see below), but the responsibility for most DC prisoners was turned over to the Federal government some years ago. The Federal government places DC prisoners in their Federal prison system, spread across the United States. It is not at all unusual for people convicted of crimes in DC to be placed in a prison in Oregon, for instance.
The following is from the DC Department of Corrections website:
"The Central Detention Facility (CDF/DC Jail) is located in Southeast DC at 1901 D Street, SE. The current facility was opened in 1976. In July 2002, the United States District Court for the District of Columbia vacated the population capacity limit at the DC Jail imposed in the 1985 Campbell v. McGruder court ruling. In October 2007, the Executive Administration established a population capacity at the jail of 2,164. This number is within the ranges recommended by two consultants hired independently by the Council and the Department of Corrections leadership during 2004 and supports the agency’s compliance with the Jail Improvement Act of 2003. The majority of male inmates (females are housed at the Correctional Treatment Facility) housed in the Central Detention Facility are awaiting adjudication of cases or are sentenced for misdemeanor offenses. As a result of the National Capital Revitalization and Self-Government Improvement Act of 1997, sentenced felons are transferred to the Federal Bureau of Prisons."
In short, if you've been convicted of a serious crime in DC, you're sent to the Federal prison system. Only folks awaiting trial or sentencing, or convicted of a minor crime, are housed in the DC Correctional Facility.
Since I wasn't fully aware of all of the above until you asked your question, I'll give you a star for posting an interesting question.