The gemara discusses the din of כבוס בגדים, meaning that if blood from the korban chatas gets onto the bigdei kehuna they must be washed. The mishna 93a says that the din of kibus must be done "b'makom kadosh" meaning inside the azara (courtyard) of the beis hamikdash, which is written clearly in the pasuk תכבס במקום קדוש (vayikra 6:20). The mishna on 94b says that a garment that had blood of a chatas on it that left the azara must be brought back in to be washed, but if it became tamei so that it can't be brought back in to be washed (because tu'mah can't be brought into the azara) it should be torn to remove the tu'mah status, then bring it in to wash it in the azara. R. Meir Simcha in Meshech Chochma quotes an interesting question from his son in law (R. Avrohom Loftiver who was married to his daughter for a short time, both R. Meir Simcha's son in law and daughter died in his lifetime leaving behind no descendants. The son in law of the Ohr Sameiach was very close with R. Menachem Ziemba and that is why R. Menachem Ziemba named one of his seform zerah avrohom after him because he died childless - see here). Since the Rambam (Beis HaBechira 6:8) paskens that the chambers which were built outside the azara but opened up to the azara had status of azara for eating kodshei kodshim, but not for schechting kodshim kalim and one who entered b'tuma was patur, there should be a solution for the garment. The garment should be brought into one of these chambers while it is tamei through a window to the outside even without tearing it first, then it should be cleaned in the azara?
The Meshech Chochma offers an answer to this question from a diyuk in the pesukim. The source that the chambers have status of azara for the purpose of eating kodshei kodshim is derived from the pasuk בחצר אוהל מועד יאכלוה - התורה ריבתא חצרות הרבה - זבחים נו ע"א. The reference to the azara as חצר אוהל מועד is the source to include the chambers. However, in the context of washing the clothes it refers to the ohel moed as במקום קדוש, which implies inside the actual azara itself, and doesn't include the chambers. As the Ohr Sameiach explains that the chambers don't qualify as the azara for the purpose of avodas ha'dam, and if the blood would be brought there it would be pasul as if it were brought out of the azara, those areas also don't qualify for the laundering במקום קדוש. The laundering must take place in the actual azara in a place that is fit for the sprinkling of the blood, to the exclusion of the chambers.