[Bruce MacCulloch]

They were a sub-race of Men.
In "Of Dwarves and Men" in Peoples of Middle Earth (HoME vol 12) it says:

Hobbits on the other hand were in nearly all respects normal Men, but of very short stature. They were called 'halflings'; but this refers to the normal height of men of Numenorean descent and of the Eldar(especially those of Noldorin descent), which appears to have been about seven of our feet. Their height at the periods concerned was usually more than three feet for men, though very few ever exceeded three foot six; women seldom exceeded three feet. They were not as numerous or variable as ordinary Men, but evidently more numerous and adaptable to different modes of life and habitat than the Drûgs, and when they are first encountered in the histories already showed divergences in colouring, stature, and build, and in their ways of life and preferences for different types of country to dwell in (see the Prologue to The Lord of the Rings). In their unrecorded past they must have been a primitive, indeed 'savage'people, but when we meet them they had (in varying degrees) acquired many arts and customs by contact with Men, and to a less extent with Dwarves and Elves. With Men of normal stature they recognized their close kinship, whereas Dwarves or Elves, whether friendly or hostile, were aliens, with whom their relations were uneasy and clouded by fear. Bilbo's statement (The Lord of the Rings,1) that the cohabitation of Big Folk and Little Folk in one settlement at Bree was peculiar and nowhere else to be found was probably true in his time (the end of the Third Age); but it would seem that actually Hobbits had liked to live with or near to Big Folk of friendly kind, who with their greater strength protected them from many dangers and enemies and other hostile Men, and received in exchange many services. For it is remarkable that the western Hobbits preserved no trace or memory of any language of their own. The language they spoke when they entered Eriador was evidently adopted from the Men of the Vales of Anduin (related to the Atani, in particular to those of the House of Beor); and after their adoption of the Common Speech they retained many words of that origin. This indicates a close association with Big Folk; though the rapid adoption of the Common Speech in Eriador shows Hobbits to have been specially adaptable in this respect. As does also the divergence of the Stoors, who had associated with Men of different sort before they came to the Shire.

In the same article, still speaking of Men:

Thus it came about that the Numenorean term Middle Men was confused in its application. Its chief test was friendliness towards the West (to Elves and to Numenoreans), but it was actually applied usually only to Men whose stature and looks were similar to those of the Numenoreans, although this most important distinction of 'friendliness' was not historically confined to peoples of one racial kind. It was a mark of all kinds of Men who were descendants of those who had abjured the Shadow of Morgoth and his servants and wandered westward to escape it -and certainly included both the races of small stature, Drûgs and Hobbits.

In Letters, 131, the Professor says

The Hobbits are, of course, meant to be abranch of the specifically human race (not Elves or Dwarves) - hence the two kinds can dwell together (as at Bree), and are called just the Big Folk and the Little Folk.They are entirely without non-human powers are represented as being more in touch with 'nature' (the soil and other living things, plants and animals), and abnormally for humans, free from ambition or greed of wealth.They are made small (little more than half human stature, but dwindling as the years pass) partly to exhibit the pettiness of man, plain unimaginative parochial man - though not with either the smallness or the savageness of Swift, and mostly to show up, in creatures of very small physical power, the amazing and unexpected heroism of ordinary men 'at a pinch'.

Letters, no 316, says:

... and that my 'hobbits' were in any case of wholly dissimilar sort, a diminutive branch of the human race.