As mentioned on the Wikia article on Paulie:
The amount of rope he carries is absurd, (which led some to speculate it was the power of a Devil Fruit) measuring hundreds, if not thousands of meters, as he was actually able to make a huge, complex rope-work in a split-second to keep not only himself, but also the entire Franky Family, their Kingbulls, Oimo, and Kashi from falling into the huge pit in Enies Lobby (as seen above). The amount of rope that he uses as a weapon and brings with himself all the time never seems to run out, since he has been known to pull out yards upon yards from time to time.
He seems to pull the ropes out of Hammerspace:
The term hammerspace describes the seemingly invisible place from which fictional characters, such as cartoon characters, pull out very large objects, such as mallets.
As for an in-universe reason/logic behind him being able to pull out so many ropes, there is none. Oda was asked this question in SBS 39:
D: Oda-sensei, hello!! Umm, where does Paulie's rope come from? And where did he come up with the names for moves like Rope Action "Half Knot"? They're not just random, are they? Tell us!
O: Paulie's ropes come shooting out of the bottom of his sleeves. Apparently, there's a whole bunch of rope hidden in the lining of his coat! [...]
Therefore, it's not that he has a Devil Fruit, but that the matter wasn't really paid attention to. It works because Oda wants it to work. There is always a possibility that he may have gotten a Devil Fruit power after the timeskip (although it's unlikely), but even if that were the case, it definitely wouldn't explain the infinite length of rope he had on him before the timeskip. So no, he doesn't have any devil fruit power and/or additional abilities which allow him to store such a length of rope on him, except for possibly wearing a coat whose lining allows him to pull out such ropes.