Your second question is really easy to answer:
Q: How do aircraft designers make landing gear that can land on a grass field without plowing into the ground and ripping off?
A: Big Fat Low-Pressure Tires.
A wide, low-pressure tire has a larger contact patch with the ground, spreading the aircraft's weight over that larger area helps prevent the tire from digging into the ground, and lets the aircraft roll more easily on rough or soft surfaces.
Your first question - How much weight can a grass strip support? is a much tougher one. The bearing strength of an unimproved surface depends on a number of factors (some big ones: The composition of the runway bed, its drainage, and what kind of surface cover - turf, rocks, etc. - it has).
If you look at aerial photos of Edwards Air Force Base you'll notice part of the runway complex includes a dry lake bed:
Some pretty heavy stuff has taken off from and landed on those runways - space shuttles have landed there.
Similarly TACA Airlines Flight 110, a Boeing 737, landed on a levee at NASA's Michoud Assembly Facility (and was subsequently repaired and took off again, though they used an access road for the takeoff rather than the grass levee).
Conversely if you tried to land a Cub on a grass strip that's saturated with water after a rainy month there's a chance your tires might sink into the mud.
CBR Technologies has a nice presentation on unpaved runways, which includes some discussion of how their bearing strength (and thus maximum landing weights, tire pressure limits, etc.) can be calculated, but each unimproved runway will be different, and the values will likely change over time.