The holiday laws of the individual federal states regulate what is allowed on Sundays, which is why they also differ depending on where you live. However, publicly noticeable activities that disturb the peace of the day are generally prohibited.
This also includes, for example, hanging up laundry in the garden or on the balcony. Although this does not make any noise, people could feel disturbed if the area is visible to third parties.
In practice, very few people will probably really be bothered if you hang out your laundry on a Sunday. After all, many employees only get to do these and similar household tasks on weekends.
Apart from that, you should make sure that your washing machine does not wash louder than the room volume on Sundays. If the device is uneven and spins at high speeds, this can disturb the neighbours. If your washing machine is in a common room, the law may also ban Sundays.