From the NHC:

HURRICANE FORCE WINDS EXTEND OUTWARD UP TO 60 MILES...95 KM...FROM THE CENTER...AND TROPICAL STORM FORCE WINDS EXTEND OUTWARD UP TO 205 MILES...335 KM.

From Stormcarib's "How Close Can It Get?" applet:

Results for Grand Cayman, Cayman Islands (19.28N, 81.35W):
The approximate Closest Point of Approach (CPA) is located near 18.4N, 81.6W or about 63.7 miles (102.4 km) from your location. The estimated time of when the center of the storm will be at that location is in about 25.3 hours (Monday, August 20 at 12:18PM EDT).

I guess that's like being strapped down to a table while someone holds a circular saw 1/4" away from your face... you're going to feel it, but you're not going to get hurt by it. Or so the thought goes. A lot of people are perkier today thinking that we might be spared completely, but the tropical storm force winds extend out 205 miles, so that's a storm diameter of 410 miles. At 18mph, that means from the time we start feeling the effect to the time that it clears up again is 22 hours at up to almost hurricane strength winds, with of course higher gusts. Storm surge is also a concern, the last update says seven to nine feet so if we're right at the edge of the hurricane wind field, we should have minimal surge, but still pretty massive wave action.

Of course, it could wobble close enough to throw the entire previous parargraph to the wind (haa, I'm funny, I know) and bring that buzzsaw ashore here in Grand Cayman.