Shakespeare wrote his plays in Early Modern English. Part of the reason it sounds somewhat archaic to the modern ear is that they were also written largely in an unrhymed iambic-pentameter poetic form.
The vocabulary Shakespeare used was extensive. Throughout his works, he used over 20,000 different English words, inventing nearly 1,700 words himself. Compare this to the King James Bible or a modern major newspaper, each of which uses about 8,000 words.
In addition to vocabulary, the challenge of reading Shakespeare's work comes from his complex sentences. Part of this was due to the iambic-pentameter rhythm, which makes it easier for actors to memorize their lines and for audiences to understand them when spoken but harder for readers to comprehend on paper.