Skip the traffic and not so cleanliness and outdoor (buggy) seating and head just 4 miles north of Boston to Medford and MOULTONS SEAFOOD... can't find fresher or more economical seafood anywhere.
178 WInthrop St
Medford, MA 02155
Moulton's Seafood Restaurant
178 Winthrop St, Medford, MA 02155
that time of year, fresh/live soft shell crabs are a must-- no comparison with the frozen ones you get the rest of the year. some good places to get them would include Peach Farm restaurant in Chinatown (either ginger-scallion, which is a bit juicier, or salt-and-pepper, which is a bit crunchier), or for pricier versions with risotto, Scampo or Prezza.
4 Tyler St, Boston, MA 02111
24 Fleet St., Boston, MA 02113
215 Charles Street, Boston, MA 02114
I actually just replied to a similar post! So here was my answer, cut and pasted here - closer to town (in Cambridge) than North Shore or South Shore:
For a take-out counter kind of a place that serves some of the best lobster rolls, fish and chips, and whole-belly fried clams in town, try Court House Seafood in Cambridge. The fish and chips and clams are very lightly battered and are sooo good! Kitchen is spotlessly clean! Three brothers run a fish market next door (Court House Fish Market) and they run this take-out counter too. They go to the fish piers daily themselves. You won't find better in town!
If you want to splurge a little, a Boston classic is Anthony's Pier 4. The Athanas family has run this place for decades and is a Boston institution - with all its quirks, fans (and some people who don't like the place too I'm sure) - but they have a great seafood menu with all the classic Boston options you can imagine - everything from New England clambakes, to lobsters served every which way you can imagine.
Court House Seafood Restaurant
498 Cambridge St, Cambridge, MA
Anthony's Pier 4 Restaurant
140 Northern Ave, Boston, MA 02210
Yeah, what you're actually looking for is more the quintessential North Shore/South Shore food experience. There aren't any actual clam shacks in Boston itself. (Jasper White's doesn't count. It's not a clam shack, but a kinda kitschy/touristy recreation of a clam shack. Clamshackmania, if you will.)
A few minutes south of Boston, you have a few shacks, Tonys and the Clam Box on Wollaston Beach; for us locals we love it. I'm sure the North Shore clan will chime in on all the great options there. Boston proper, doesn't have much other than the Barking Crab and it doesn't sound like what you are looking for. If you are going to be here for a bit and have a car you may want to head north to coastal Maine and stop at one of the great lobster shacks there.
The Barking Crab
88 Sleeper Street, Boston, MA 02210
789 Quincy Shore Dr, Quincy, MA 02170