This appears to be an an area of some debate. There are five cities that each claim to be the oldest continually inhabited cities:
Damascus (and the Barada basin) claims to date back to around 9000 BC, but the first evidence of large scale settlement is found around the second millennium BC.[Source]
Jericho also claims an origin around 9000 BC, but adds they were the first walled city with the first fortifications built around 6800 BC. However, they may fail the "continuous civilization" test, as they may have been destroyed, abandonded, and later rebuilt.[Source]
Byblos was settled during the Neolithic period. Carbon dating tests places the earliest settlement around the 3rd millennium BC. [Source]
Aleppo and Balkh's claims, from what I can gather, do not hold water in comparison to the others. Their civilization dates to the Chalcolithic period, around 4300 BC. [Source].
I'm sure that I overlooked some things in creating this, so please correct me if you see any inaccuracies. Wikipedia's list-style entry could use some cleaning up, but is a fairly good reference if you have some time to scan through them all and investigate.
I suggest that the oldest still inhabited settlement is unknowable.
All you can ask for is the oldest archaeological example known.
Göbekli Tepe and connected village wins (Edessa/Urfa) - at around 11,000+ years ago. Radiocarbon dating as well as comparative, stylistic analysis indicate that it is the oldest religious site yet discovered anywhere.
But if it was uninhabited for 20 years in 200-300BC - does it count? 50 years? 100 years? What counts?
There are regions in the Kimberly's in Australia that are still currently inhabited, where stone art depicts animals extinct since the before the Mesolithic 20–18,000 BC. Radiocarbon dating in these extremes is problematic. Such depictions of living animals - found only in the fossil record - is not uncommon in Australia. Aboriginal people can still live right by and even regularly visit the same communal fire-pits, in many cases. Digging into the local middens will send an archaeologist back into the mists of time - uncorroborated by concurrent, comparable culture and known records - and in some cases - even known species.
The San Marcos River banks, now San Marcos, TX is considered one of the oldest continuously inhabited sites in the northern hemisphere. "Archeologists have found evidence at the San Marcos River associated with the Clovis culture, which suggests that the river has been the site of human habitation for more than 10,000 years" (wiki, the city of San Marcos, & Texas state university) while it is not the same settlement, it is a place of continual settlement. While there are places which probably are longer in the Middle East and Africa it is interesting that a place in Texas can boast one of the longest records for continual settlement.
I have a question myself. Would Mexico City be considered one of these still inhabited settlements like the others in the thread.
Here is evidence of occupation at London going back to 4500 B.C., not sure if it is contiguous or not though... http://www.thamesdiscovery.org/frog-blog/london-s-oldest-find-discovered-at-vauxhall
/r/askanthropology might have some good answers for this question.
Here's a list of some highlights: http://listverse.com/2012/12/01/10-ancient-cities-still-around-today/