?matplot often convenient for this kind of task when building the R base:
matplot( dat$Age, dat$CO2 + outer(dat$Standard_error, c(0,1,-1)), type="l", lty=c(1,2,2), col=c(1,2,2), xlab="Age", ylab="CO2" )

If shading is critical, I'll move on to polygon :
ses <- dat$CO2 + outer(dat$Standard_error, c(1,-1)) with(dat, plot( Age, CO2, type="l", ylim=range(ses), panel.first=polygon(c(Age,rev(Age)), c(ses[,1],rev(ses[,2])),border=NA, col="#ebebeb") ) )

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