I'm kinda hermity and mostly prefer to not escape the house. When I feel stir crazy, I go out by myself for a long brisk walk while listening to music. My usual route loop is about 2 miles, but sometimes I go 5-6 if I'm super antsy.
There are some at-home jobs that require a set time frame, have require hours spent per week, need there to be no background noise, etc. To be fair, you're more likely to encounter this kind of job. In this situation, you can not care for a baby/toddler and work at the same time.
I've worked three very different types of jobs at home, all of them with kids underfoot. I started working at home after the birth of my first child. It was my employer's idea! She knew I liked my job, but she was also a fan of kids having a SAH parent. She hired a person to handle things that had to be done in person at the office, and I did all the rest from my home office while caring for an infant; later an infant and a toddler. I did that for years, until the business closed with the owner's retirement.
For a couple years I worked for an internet company. I had no set hours and no human interaction. That job wasn't 'living wage' it was $300-$500/mo depending on how much time I put in.
Now I am self employed providing a service. My kids are in school now, so it isn't a problem to have meetings during the day or be on the phone. I can still do the rest of the work whenever, it doesn't matter if they are home or not.
Gamma G's comment about not paying taxes if you didn't have a 'real' job brings up a good topic - Retirement Planning.
Even when you don't have a job with an outside employer, do properly claim your income. Even if your income is low, sporadic, or from multiple sources.
#1 - You get 1 Social Security credit for earning $1,220. You can earn 4 credits per year. Anyone that earns just $4880 in a year gets the max SS credit, even if their income is too low to pay the other income taxes. To qualify for full SS benefits later you only need to accumulate 40 credits over your lifetime.
#2 - If you're single, you need to have reported earned income in order to save for retirement in tax-advantaged accounts like IRAs and Roth IRAs.
If you're married, you don't need to have any personal income at all and you can still have your own retirement account. So if you're pure unemployed SAHM, you should still look into opening an account for yourself.
*Amounts and laws that are current as of the date of this posting.*