"Scylla and Charybdis" takes place in the National Library and includes both Stephen (holding court amid a group of Dublin literati) and Bloom. What Joyce does here with the Homeric correspondence is quite remarkable. You may recall this Homeric episode: Ulysses must sail his ship safely between Scylla (a six-headed beast) and Charybdis (a huge whirlpool). Ulysses stays close to Scylla, losing six of his men, but later must return and loses his entire ship to Charybdis. Similarly, both Stephen and Bloom must navigate a "middle path" between two dangerous extremes. Recall their journeys thus far: Stephen has come to the library from the newspaper office, Bloom from lunch. Stephen must mediate between his Scylla - the crass, reductive world of newspaper reporting - and his Charybdis - the lofty, self-indulgent world of intellect and ideas (the Irish Revivalists of this chapter). In other words, Aristotle's actualities and Plato's abstractions. Bloom must mediate between his Scylla - the world of pure physicality and sensation (good ol' Buck Mulligan!) and intellect (Stephen). Notably, he passes between them upon exiting towards the end of the chapter.
Much of the chapter is dominated by Stephen's somewhat difficult "theory" with regard to Shakespeare: in short, Stephen theorizes that in playing the role of the Ghost in Hamlet, Shakespeare is speaking to both the "son of his soul" (Hamlet) and the "son of his body" (his deceased son Hamnet) about his (Shakespeare's) own wife's infidelities. Note that Odysseus, Bloom, and Shakespeare/Hamlet all go on journeys, return, and face the prospect of a wife being usurped. (If you really want to make your head spin, go ahead and throw Joyce's own biography into the mix!) For Stephen, Shakespeare - "the father of his race" (208) - is the consummate artist, both in terms of giving "birth" to literary/artistic creations (no female required!) and in merging the subjective with the objective: "He found in the world without as actual what was in his world within as possible" (213). In other words, Shakespeare has reconciled the swirling whirlpool of inner thought and subjectivity that threatens to swallow us up (Charybdis) with the cold, hard, objective facts of the external world (Scylla).
One can't help but think that Joyce is speaking here of his own literary life as well. Like Stephen, Joyce successfully escapes the backward-looking Romanticism of the Irish Revivalists. The comment "Our national epic has yet to be written" (192) seems to imply two questions: Is Stephen the right man for the job? and Is Joyce's Ulysses that book? Finally, Stephen's great comment on Shakespeare sounds an awful lot like an assessment of Ulysses itself: "Every life is many days, day after day. We walk through ourselves, meeting robbers, ghosts, giants, old men, young men, wives, widows, brothers-in-law. But always meeting ourselves" (213).