Museums must prepare for a post-digital era in which the cultural record will increasingly disappear without leaving stable objects behind. Contemporary history and cultural anthropology are now written through digital infrastructures, yet born-digital heritage is unstable, uneven, and often lost before its value can be recognised. Web archives aim to combat this forgetting, but they operate in a digital realm where what survives is partial and transformed through capture. This raises a museum question as much as an archival one: how can museum’s exhibit cultural memory when the object itself may have disappeared and never been tangible? Using the Digital Ghosts exhibition as an example, this article explores how museums can make digital decay visible, spatial, tactile, and affective. The exhibition translated web archive metadata into interactive visualisations, data physicalisation, and embodied encounters with decay and uncertainty. As the lifecycles and nature of cultural objects have changed, innovation in museums is not just a matter of superficial immersion or modernisation. Museums will need to move beyond valuing only what survives and develop methods for exhibiting what preservation misses, what disappearance feels like, and why absence itself has become part of cultural heritage.