Personally, I think that most "legal" (for lack of a better word at this late hour) agencies - insurers, courts, arbitors, etc. - would probably pick an age at which most children grow up, and use that as the standard. I.e., under Anglo-Saxon common law a 10-year-old was considered capable of consent in any relationship other than those involving military service. I think we'd see something like that: 12- (to pick a number that makes sense to me) year-olds are automatically considered adults. A person younger than that could prove his or her adulthood before a Juvenile Competency Judge, of course - the "age of majority" would simply be the time at which it occurs automatically.
Of course such a system would not be perfect - suppose a 10-year-old ran away from home and went to get a job, and finds that nearly every business owner asks to get permission from his mom and dad before hiring them, for liability reasons. This would mean that either he would have to get a juvenile competency hearing, work for a less reputable company, or go back home. So it wouldn't necessarily be a Rothbardian utopia wherein self-emancipation would happen in a matter of seconds. If the society was paternalistic, the standard age of majority may be even higher, and judges may be unwilling to declare adulthood upon prepubescents, no matter how bright.
But I don't think that's likely to be the case. Without a government-enforced age of majority, society would basically find a market-based age of majority which may in fact be quite low. If 8-year-olds were constantly going around looking for work and proving themselves capable, autonomous members of society, then that would be the prevailing age of majority. I think, however, that it would probably fall somewhere around puberty.
In some cases the parents might protest that the young person is still a child, but the burden of proof would be on them to show that to be the case.
In cases of abuse, naturally children would be treated differently. A child would be free to change guardians without some market-based form (or shadow of a form) of CPS getting involved, although such an industry might exist, to whom a person could go who suspects abuse against a child.