Hep c it transferred by blood to blood contact so if they were cut by something and bleed and you later were cut by the same sharp edge and bleed there would be a very small risk. For example in the situation where a healthcare worker experiences an accidental needlestick involving a patient with known hepatitis C the risk of transmission is about 1.8% .
Washing the clothes does not result in blood to blood contact of the hepatitis C virus. So not a risk.
https://www.cdc.gov/hepatitis/hcv/cfaq.htm#cFAQ34
"How is Hepatitis C spread?
Hepatitis C is usually spread when blood from a person infected with the Hepatitis C virus enters the body of someone who is not infected. Today, most people become infected with the Hepatitis C virus by sharing needles or other equipment to inject drugs. Before 1992, when widespread screening of the blood supply began in the United States, Hepatitis C was also commonly spread through blood transfusions and organ transplants.
People can become infected with the Hepatitis C virus during such activities as
Sharing needles, syringes, or other equipment to inject drugs
Needlestick injuries in health care settings
Being born to a mother who has Hepatitis C
Less commonly, a person can also get Hepatitis C virus infection through
Sharing personal care items that may have come in contact with another person’s blood, such as razors or toothbrushes
Having sexual contact with a person infected with the Hepatitis C virus"
Note: Hep c is not generally considered to be an STD. In the case of long term monogamous couples the CDC does not recommend the use of barrier protection i.e. condoms.
For those who have multiple sexual partners, engaging rough sex practices, or the presence of the HIV there is an increased risk of transmission.
"Can Hepatitis C be spread within a household?
Yes, but this does not occur very often. If Hepatitis C virus is spread within a household, it is most likely a result of direct, through-the-skin exposure to the blood of an infected household member."
"What are ways Hepatitis C is not spread?
Hepatitis C virus is not spread by sharing eating utensils, breastfeeding, hugging, kissing, holding hands, coughing, or sneezing. It is also not spread through food or water."