In 1873 Jules Verne wrote the novel "Around the World in Eighty Days". Is actually the least imaginery of Verne's books and was completely doable with the means of the era.
Nowdays, assuming that someone wouldn't use airplanes which are a bit unfair (there are
direct flights Paris to Tokyo, Tokyo New York and New York to Paris. I mean.
Come on) and would only stick to surface transportation and ships like Verne's character, someone could do it today in about 42 days and it would look something like this:

Apparently the longest piece of the journey is the ship from Japan to Canada (13 days, ouch).