Can a person's individual carbon emissions become negative over their lifetime?
Aussies emit ~15 tonnes CO2 per capita per year. Current carbon capture and storage technology can bury CO2 for \$40-\$57 per tonne (estimate for early 2020s). So, yes, you could do this for ~$50,000 .
Underground storage is relatively permanent. As for trees, you could plant one of these,

1,700 tonnes  CO2 and lives 3,000 years. With any kind of trees you will need to assure your patch of forest remains treed for millenia. As for the number of trees, I think forested area is a better measure:


Approximative values of the carbon content (tonnes per hectare) for various vegetation types.
Source : IPCC, 2001

You would need tens of hectares.