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Magic seems to accompany the other races as they fade; while both Sauron and the elves use magia and goeteia, they, as "mythological" figures, are bound to dissappear. The dwarves seem to have magic abilities (judging from their song in the Hobbit, their participation in the magic protection of the troll hoard, their moon letters and the magic doors of Moria).
Concerning the race of Men, Tolkien states in letter #155 that magic "is an inherent power not possessed or attainable by Men as such"; the only exceptions found to this are the swords of the Westernesse "wound about with spells for the bane of Mordor" and the healing power of Aragorn (but in both cases there is an "elven descendancy" element involved).
The hobbits are a branch of Men, so it is rather unlikely they have magic powers (more or less seriously, Tolkien notes in the first chapter of the hobbit that "there is little or no magic about them, except the ordinary everyday sort which helps them to disappear quietly and quickly when large stupid folk like you and me come blundering along, making a noise like elephants which they can hear a mile off").
In the fourth age, the presence of magic among Men is bound to be restricted to the use of whatever magical objects are left (glowing swords, elven ropes cloaks and boats, the palantiri and to a much lesser extent Galdadriel's blessed earth given to Sam).
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