Heat a 4-qt. pot over high heat. Add mussels and 1/4 cup water; immediately cover to trap steam. Cook just until mussels open, 2 to 4 minutes. Drain, discarding any unopened ones. Let cool, then scrape mussels from shells into a bowl.
Heat oil in a medium pot over medium heat. Add garlic, shallot, paprika, and saffron. Reduce heat to medium-low and cook, stirring often, until garlic and shallot are soft but not golden.
Remove mixture from heat and stir in salt and vinegar. Pour over mussels. Set bowl in a larger bowl of ice and cold water and let mussels cool until cold.
Transfer mussels and pickling mixture to a pt.-size jar (or 2 smaller ones). Tamp mussels gently to just submerge, then chill at least 1 and up to 3 days. Bring to room temperature before eating.
*Find the paprika at well-stocked grocery stores and online at spanishtable.com.
Make ahead: Up to 3 days, chilled.