There are several possible ways the Hobbits could have disappeared from the view of Big Folk.
Firstly, it could simply be as has been said that they reintegrated back into the larger human population either by means of evolution or by intermarriage. Yes, Aragorn may have set a ban on Big Folk entering The Shire, but Aragorn did not live forever, nor would all his descendants have been so scrupulous.
It is also possible that Hobbits simply left The Shire; presuming that those who left were the more enterprising and adventurous, that would leave only the parochially minded back home, and the society could have declined. That gives food for thought for the Big Folk on t'other side of the pond.
Or it could be that Aragorn's ban on Big Folk entering The Shire actually did cause the Hobbits to disappear from our perception and history. In this way they may have passed back into the realms of myth and legend and been forgotten, even if they did still (and do still?) exist. In some historic cases of aboriginal peoples being put into reservations ostensibly to protect them, this move has actually destroyed their cultures; the young see the poverty around them and instead choose to leave for the wider world and a life of opportunity, thus even further hastening the decline of those traditional cultures.
If you want to be pessimistic there are a lot of reasons why the Hobbits may simply have grown extinct. Yes, that seed was sown which may have destroyed their innocence. They did indeed have battles before the time of the WotR, but these were battles within The Shire, not events from that war brought into their little world. After Saruman and Frodo (because Frodo and Sam and Merry and Pippin also brought the outside world into the Shire with its 'fancy foreign talk and fancy foreign ways') would Hobbits ever be the same? No longer a parochial race, globalisation (trading pipeweed for Lembas like trading coffee beans for Coke?) had hit this corner of Middle-earth. It wouldn't be long before they succumbed to a more homogenised Middle-earth.
There is also the more poetic explanation that the world of The Hobbits simply disappeared from our sight. That idea has been inspired by reading Mists of Avalaon, where the world of the druids and descendants of Atlantis did not get wiped out but remains hidden behind the mists and is ready to find for those who wish to find it. That would mean The Shire is now in Faerie. In Mists of Avalon and a whole lot of other literature there are many references to little people who may or may not be remnants of older cultures, Picts, the Fir Bolg, Boggarts, Pixies etc. Tolkien was only building on the myth and legend of older peoples as many before him have done when he created The Hobbits. So from that point of view, the Hobbits are still there, are still living in The Shire, but you will only find it if you really really want to find it. You might catch a fleeting glimpse of a Hobbit out of the corner of your eye at the end of the garden, amongst the trees or you might smell tobacco wafting mysteriously from somewhere and not see a smoker. That might be a Hobbit.
I like the last one best but all those ideas are possible.