It was possible, but not easy. The difficulty of the task accounts for the substantial amount of the bet: £20,000 in 1873 is worth about £2,000,000 or more than US $2.5 million today.
William Butcher's 1995 translation of Verne's book includes an appendix that provides details of contemporary sources that had information regarding quick circumnavigations, both theoretical and actual. A periodical called Le Tour du Monde carried an article on the occasion of the opening of the Suez Canal in 1869, when I was but a wee lad. The article said that it was now possible to complete the circumnavigation in 80 days, with the following itinerary:
Paris to Port Said, railway and steamer............6 days
Port Said to Bombay, steamer ......................14 days
Bombay to Calcutta, railway .......................3 days
Calcutta to Hong Kong, steamer ....................12 days
Hong Kong to Edo, steamer .........................6 days
Edo to Sandwich Islands, steamer ..................14 days
Sandwich Islands to San Francisco, steamer.........7 days
San Francisco to New York, railway.................7 days
New York to Paris, steamer.........................11 days
Total ............................................ 80 days
(Port Said is the head of the Suez Canal; the Sandwich Islands are Hawaii. ) This theoretical journey is pretty much the exact itinerary proposed by Fogg, so Verne must have seen this article and found it an inspiration for his novel.
Another inspiration was the American entrepreneur George Francis Train. In 1870, he traveled around the world in 80 days. Well, the actual travel took 80 days, but he broke his journey and hung around in France for a while. From the Wikipedia article linked above:
in 1870 Train decided to make a trip around the globe, which was covered by many newspapers. The actual traveling took 80 days, though he stayed two months in France, supporting the Paris Commune for which he spent two weeks in jail (the US government and Alexandre Dumas intervened to get him released). His exploits possibly inspired Jules Verne's novel Around the World in Eighty Days. His protagonist Phileas Fogg is believed to have been partially modeled on Train.
One remarkable person to complete a circumnavigation in less than Verne's stated time was the investigative journalist Nellie Bly. Following the publication of Verne's novel, she decided to try to see whether the journey could actually be completed in 80 days. In 1889, she went around the world in 72 days.