The Interplanetary Filesystem (IPFS) is a distributed data storage service frequently used by blockchain applications and for sharing content in a censorship-resistant manner. Data is distributed within an open set of peers using a Kademlia-based distributed hash table (DHT). In this paper, we study the structure of the resulting overlay network, as it significantly influences the robustness and performance of IPFS. We monitor and systematically crawl IPFS' DHT towards mapping the IPFS overlay network. Our measurements found an average of 44474 nodes at every given time. At least 52.19% of these reside behind a NAT and are not reachable from the outside, suggesting that a large share of the network is operated by private individuals on an as-needed basis. Based on our measurements and our analysis of the IPFS code, we conclude that the topology of the IPFS network is, in its current state, closer to an unstructured overlay network than it is to a classical DHT. While such a structure has benefits for robustness and the resistance against Sybil attacks, it leaves room for improvement in terms of performance and query privacy.