Can someone else claim my children this year since I didn't have a job?

If the other person provided over half of their living expenses (presumably including housing) they may be able to take one or more dependency exemptions. You should contact a tax professional.

Each calendar year is an independent decision, so who deducts them this year has no effect on next year. If somebody elswe provided a majority of the support for your children then they can usually take them. If nobody provided a majority, then you can agree to let someone who provided more than 10% tqake them, so long as nobody else does. Be aware that support is only on of the 5 tests for a dependent. The dependent must not be married (or not file a joint return), must be a citizen or resident of the USA, Canada or Mexico, and must have low enough income that he or she is not required to file return. Also, the dependent must be a relative of the taxpayer or a member of the taxpayer's household for the whole year.

I would need many more details to answer your question as to this year. As for the future, the facts are not yet known. The IRS regulations rule unless there is an order of the court.