The rapid evolution of viruses generates proteins that are essential for infectivity and replication but with unknown functions, due to extreme sequence divergence. Here, using a database of 67,715 newly predicted protein structures from 4,463 eukaryotic viral species, we found that 62% of viral proteins are structurally distinct and lack homologues in the AlphaFold database. Among the remaining 38% of viral proteins, many have non-viral structural analogues that revealed surprising similarities between human pathogens and their eukaryotic hosts. Structural comparisons suggested putative functions for up to 25% of unannotated viral proteins, including those with roles in the evasion of innate immunity. In particular, RNA ligase T-like phosphodiesterases were found to resemble phage-encoded proteins that hydrolyse the host immune-activating cyclic dinucleotides 3',3'- and 2',3'-cyclic GMP-AMP (cGAMP). Experimental analysis showed that RNA ligase T homologues encoded by avian poxviruses similarly hydrolyse cGAMP, showing that RNA ligase T-mediated targeting of cGAMP is an evolutionarily conserved mechanism of immune evasion that is present in both bacteriophage and eukaryotic viruses. Together, the viral protein structural database and analyses presented here afford new opportunities to identify mechanisms of virus-host interactions that are common across the virome.