Sounds entirely feasible to me. We've been doing a ~2k/mo budget (or less, or WAY less, in varying situations) for years now. Our kids are 6 & 7 now, and if they're going to be in school, you can always work for some extra spending money during school hours.
If you pay attention, you'll probably find that your "household supplies" budget line will end up more than $50/month on average, while your gas/insurance may well be far less, especially if you bike most places (we have a '96 Civic as our grocery-getter, and I fill up about once per month, which is $20-25.). Also, if you shop around, you can find internet for less than $50 (we pay $40 for a low-midrange plan with Comcast). And if you're willing/able to do a third-party cell provider like MintSim, you could have two lines for $30/month (T-Mobile network).
For instance, here is our budget, laid out by line item. Note that we own our home (well, mortgaged, but still, we pay taxes/insurance/etc plus all maintenance) and we have THREE vehicles (work truck, grocery-getter, wife's car). We're currently on medicaid for health ins, as our income is too low to hit the subsidized exchanges, and we don't have any ongoing medical conditions.
Mortgage & Rent $1,200
Internet $40
Utilities $180
Home Supplies $50
Home Phone $30
Clothing $50
Tools/Equipment $10
Personal Care $25
School Supplies $17
Food & Dining $200
Birthday Presents $20
Amusement $20
Life Insurance $17
License/Registration$16
Gas & Fuel $75
Auto Insurance $117
Service & Parts $30
Total = $2097