It's possible and sometimes quite good. With a boat you can access a lot of spots, though I never did. If you want to get pumped, check out the Dutch NDB webcam. It shows the point break that's off the North end of the runway. There's a locked gate that blocks the road to the point, but you can get a key (for a fee) from the Department of Public Safety. It's a shooting range.

Here's a link to the cam:

http://akweathercams.faa.gov/sitelist.php
From there, select Dutch NDB, then North.

The other good spot to surf is Summer Bay. It's a beach break. There's a reef break just beyond summer bay that looked good on occasion, but I never tried it because it's beyond my skill level. Here's a few pics/videos I took of the reef breaks one day:

http://gdwilco.blogspot.com/2009/03/for-justin.html

Occasionally you can surf the spit, but it has to be pretty big to work. Every once in a while, if it's big, it can provide a very long left. But that's rare. Mostly, if it's working you'll get 50-100 yards or so.

I was the only person out there with boards when I left about 8 months ago. You'll have to bring your own. A 5/4 with a hood will keep you plenty warm.

FYI The best spots in Alaska to access surf by road are Yakutat and Kodiak, Sitka sometimes, Dutch sometimes, Homer sometimes, Seward (rarely) and possibly a few other places. Cordova requires a boat or a plane.