It's pretty hard to be thrifty as a family in Tokyo, although it is possible in regional cities and in rural areas. It is generally accepted that the minimum a 2-child Japanese family can get by on, if mom is staying at home, is about JPY4.5M a year. This includes everything and assumes that the family is renting, but possibly saving a small amount each month for a mortgage on a house by the time the parents are in their late 30's or early 40's.
Mom watches for food specials, rides a bicycle to do shopping, and keeps the family to a fairly Spartan Japanese seasonal diet. The kids go to the local school, and rely on public or school facilities for entertainment. Dad takes a 60- to 90-minute commute to get to work - luckily; the company pays the commuting cost. The rent deposit for their apartment was probably loaned by the parents and mom probably gets some help from the in-laws to look after the kids when she needs to go out - so no baby sitter.
As a foreign family, when would you want to be on this rather thrifty level of living? The most common instance is where the foreign spouse is accompanying his/her Japanese partner back to Japan because the partner's parents are getting old, or because both parents have decided that the kids need to establish some Japanese roots. Families in these situations are not moving for the money, and will put up with a lot more difficulties than the average foreign family. In a way, the hard times will become "the good old days" as they age and the financial situation improves.
I'll note that if you fall into the category above, you can be even thriftier by spending 6-12 months living with your Japanese in-laws. As I've related in previous articles, this is not always an enlightening experience, but it WILL be excellent for your Japanese speaking practice and also for saving the initial deposits needed to move in to your own place.
A. One-time costs:
1. Shopping bicycle - JPY20,000
2. Apartment deposit, 60-70 sq. m. 2DK about 90 minutes away from center of city (deposit includes 5 months rent and 1 month in advance) - JPY720,000
3. Basic furniture for 4 bought from Japanese mail-order catalog, mainly consisting of futons, clothes chest, kotatsu, kitchen table and chairs, kitchen utensils, and 2 desks for the kids - JPY300,000
4. Standard Japanese school, uniforms, bags, etc. - JPY100,000 x 2
1. Apartment rent, Chiba or Kawasaki - JPY120K/month
2. Food for 4, few or no foreign items - JPY100K/month
3. Bento for Dad, videos, family restaurant once a month, no baby sitter - JPY40K/month
4. Clothes, utilities, doctor visits, misc. - JPY80K/month
5. Kids' activities - JPY30K/month
TOTAL: JPY370K/month => JPY4.44M