Question

How do cells repair themselves

Answer 1

They have a function "Mitosis"

Answer 2

You're asking how do cells repair their components once they are damaged. There are a variety of DNA repair mechanisms in the cell: non-homologous end-joining, homologous recombination, nucleotide excision repair, mismatch repair and a few other minor ones. They all function differently but the idea is that "damaged" DNA is removed and then resynthesized. As far as proteins go, cells usually just tag them for destruction using poly-ubiquitin chains that send the proteins to proteasomes (protein degradation system). Small damage to membranes is not troublesome, but lots of membrane damage, as can occur with oxidative damage, is quite devastating and can induce necrotic (cell death) response (not the same thing as apoptotic response even though they both end in cell death). I hope this helps.