New York City
renting a house in Nantucket for a week, flexible budget, want to walk to the beach, will have a car, 2 couples (65 yrs old and 38 yrs old + 2 year old)...we all love the beach and would like to be in walking distance but are wonder if we should also be close to town given we will have a few dinner there and go shopping a few times.
I've never rented a house on Nantucket before, as I've been lucky enough several times to be a guest of others who either own or rent homes on Nantucket. Thus, while I can't really speak to prices/best places to go to find homes to rent, I can give you my quick take of some of the areas on the island.
Town is fun to walk around, but it also seems to be hotter (hold the heat more) than the rest of the island. If you do stay in town, try to get a place with air conditioning. It can get steamy in the summer months, particularly if you're in a place tucked in next to a bunch of other buildings. You won't get much breeze and it can be pretty uncomfortable.
If you want to be walking distance to town but have a bit more space (where homes aren't crowded next to each other), consider going north out of town -- up around the Brant Point area (it's a pretty short and flat walk into town).
Since you will have a car, though, you might look to either find a place more centrally located towards the middle of the island, or pick a place near one of the beaches so at least you can walk to one of them.
The beaches on the north side of the island (e.g., Steps Beach, Jetties Beach) are usually pretty calm (hardly any waves) as they are protected. You can wade out pretty far and it's great for young kids. (There's also a beach right in town that is really mellow, as it's basically a part of the harbor.)
The beaches on the south side of the island (from Madaket all the way to Surfside Beach) are where the surfers go. The surf can get big and the water can be pretty rough. There also is a bit of a drop-off after you wade in a little ways. Might not be ideal for a 2-year-old.
Likewise for the east part of the island. While the waves can be small, there can be a pretty big drop-off after you get a few feet into the water.
Madaket (the southwest corner of the island) is very chill and has a good restaurant.
Cisco Brewers has the brewery (great outdoor space and often live music) and is also very chill. Surfside is nice but it's near the airport so you'll hear planes overhead throughout the day.
Oh, one last thing: Nantucket has lots of great walking/running/bike paths all over the island. You can bring bikes on the ferry or rent in town.
Hope you have fun!
Mentioned in this answer: