The events in The Hobbit and LOTR are supposed to have taken place in a forgotten pre-historical period of our own world (Tolkien stated several times that Middle-Earth was not to be understood as another planet, an alternate dimension etc. as some people thought). Hence Tolkien himself later admitted that many elements of Hobbit society were anachronistic, since they did not exist before the modern era - not only Bilbo’s clock but also his waistcoat with the brass buttons; umbrellas, silver spoons etc. In fact, Hobbit society is modeled very closely on the type of rural English culture which Tolkien knew in his childhood. Take away the curly-haired feet and so forth, and the Hobbits are essentially Edwardian Englishmen.
Tom Shippey has a good discussion of this in his wonderful book The Road to Middle Earth. He points out other, more subtle, ways in which Bilbo and the hobbits are anachronistic. For one thing, the fact that they have a daily mail service - in the first chapter it refers to Bilbo reading his daily letters - something which in real history, never existed before the 19th century.
Another example, the comical conversation between Bilbo and Gandalf at the very beginning - I never understood this until I read Shippey’s book. Bilbo uses modern English conversational expressions in his speech, and Gandalf seems never to have heard them, and takes them literally, which leads to misunderstandings. Specifically the phrase “Good morning” used as a way to end a conversation (a very British thing, it has to be spoken with a particular intonation) and the expression “I beg your pardon” as a way of expressing surprise and disbelief.
Although Tolkien himself later said that the anachronisms of Hobbit life (or at least the more obvious ones) were a mistake, I think it works brilliantly. The Hobbits are easy for us to relate to because they exhibit modern feelings and attitudes. As a result, we are able to follow Bilbo and Frodo through their adventures and we can experience all the strange and wonderful things that happen in Middle-Earth through their eyes, so to speak. We understand how the hobbits feel when they encounter Dwarves, Elves, heroic Men with swords, Wizards, Orcs and so forth, because they feel the same way that we would feel in their situation.
