When in the museum computer room, Hannah, played by Kristin Kreuk, notices Agent Shaw, played by Brandon Routh, on the monitors and asks Chuck what his ex-girlfriend is doing there with a ridiculously good-looking date. Chuck replies with "Ridiculously good, him? Yeah, if you're into that strong kind of Superman type of a guy." Routh played Clark Kent / Superman in Superman Returns (2006), while Kreuk played Lana Lang, Clark Kent's sometime girlfriend, in Smallville (2001).
The outside of the museum is The House of the Book on the Brandeis-Bardin Campus of the American Jewish University in Simi Valley, CA. It has been used for at least a dozen movies and TV shows, including Mighty Morphin Power Rangers (1993) and Red All Over (2010).
The setting for the Mask of Alexander exhibit is the "MAXMIN Museum of Art" (with this name visible on several posters). Jody Maxmin is a beloved professor of Art History and Classics at Stanford, which writer Phil Klemmer attended. (And the truck parked outside the museum is marked "Klemmer Fine Art Moving").
The mask of Alexander is not a real artifact, but it is based on the Mask of Agamemnon, an actual gold funeral mask from Ancient Greece, on display in the National Archaeological Museum of Athens.