I don't live there but recently visited. My friends who do live there spent over $1m to buy a small 3-bed house. So rents are also going to be very high. Let's say $5k a month including property tax and maybe insurance.
Health insurance - unless your company subsidises very very heavily you're going to be looking at a cost of $5k to $20k per year in premiums for a family (we pay $15k right now).
On top of that you'll have your out of pocket costs - I'd allow about $3-6k a year unless you have a great policy or are never ever sick.
You can probably lease a car for about $300 a month, then insurance and tax another $100 a month. You'll probably want a car each.
How old are your kids? You'll probably either have daycare costs, or if they're at school then remember summer holidays are 11-12 weeks so you may need childcare for the summer. I used to allow about $7-8k for 3 kids for my summer childcare cost.
Food, bills (elec and gas etc tend to be more here for some reason), internet will be about $60 a month, cable (if you want it) $120+ a month. Cellphones probably $100 a month for your first line, then $40 for your second.
Add that up, add on the cost of your taxes (potentially FICA, federal, state, city - Cali is a high tax state), private pension, life insurance, disability insurance. Then having some fun, travel, skiing, all the things you actually want to do while you're in the Bay Area.
There's your cost of living which should help you work out if $120k is enough - I would guess absolutely not.
And yes, the traffic is AWFUL.