As noted above, private rental agencies like Bermuda Rentals, Bermuda Getaway will provide you with a broad cross section of the sort of self-catering accommodations that you're looking for. What you see and read in property descriptions tends to be what you get (and usually much better). You might also check out the Bermuda listings of--
HOME AWAY-- www.homeaway.com/search/keywords:Bermuda
Bermuda is a relatively tiny place--approx. 21 square miles of landmass...21 miles from end-to-end...3-4 miles from shore to shore...No where is very far from anywhere else in Bermuda and getting from here-to-there is almost always a pleasure passing through Bermuda's incredible beauty and charms.
Bermuda's best beaches will be located out along her magnificent south shore, in Paget, Warwick and Southampton parishes. There are a fine selection of restaurants in this area but restaurants within comfortable "..walking distance.." of these locations will be limited.
Bermuda's largest concentration of restaurants (all types, cuisines, prices), shopping, nightlife (dance clubs, pubs, bars, lounges) and the nexus of her genuinely excellent public transportation system (buses & ferries) will be located in and around downtown Hamilton...but her south shore beaches will be a 15-40 minute ride from here.
With this in mind, you will probably have to make some adjustments and compromises to your, "..close to the best beaches.." and "..walking distance.." criteria...but I suspect that you'll discover that any small 'inconveniences' you may accept from one will prove to be minor and well '..worth it..' for the other.
BERMUDA--Handy Reference Map (Orientation, layout, relative locations)--
BERMUDA BUS INFO--(Routes, schedules, fares, etc)--
BERMUDA FERRY INFO--(Routes, schedules, etc)--