Am I too big for this motorcycle?
The EX250J (08-->) has a riding position which doesn't feel comfortable to some taller riders. You may want to try one out first or get a 250F if this bothers you. The stuff below applies to the 88-07 EX250F.
N250RC Grand Poobah Leon is about 6'2", and he's got over 100,000 miles on one of his 250s. Many people on 250s (Jeb/Andrew/Daryl) are around 5'10"-6'. Weight is actually the more important factor; anyone over 250 lbs. or so might want to consider the EX500 or 650R.
Jack Walshe, while seeming to have some trouble staying on the seat, isn't complaining much, and he's at least 6'4" and on the far side of 250 lbs.
The EX250 is small, but so are most sport bikes. That's the whole idea - that they're fast and lightweight, and small. Since the dealer won't let you do any sort of test ride to see how you'll fit for a long ride, sit on the bike at the dealer for about 30-45 minutes and see how you feel. Or go find a used 250 and take it on a test ride.
If you think you'll be too crowded on a sportbike, there are many dual sports and cruisers that may be more to your liking.