How did you get fired from your babysitting job? Did they move away from your neighborhood? Did they have a problem with the work you did for them? Or did they just not have the need for childcare anymore?
You could always find a similar job. Just start advertising your services. Pass out flyers, go door to door telling your neighbors the days and times you are available to work. Don't give up. There are other people out there who like to go out once in a while and may need you to watch their kids.
That's why I asked how you why you were terminated. If it was for an unavoidable reason then you could use the parents of your previous clients as a reference. They could let other parents know what a good job you did. You will get more new customers if they can talk to your old customers. So try that.