When there is no oxygen to act as a final acceptor, the chain and Krebs cycle do not function. The reduced NADs from glycolysis do need to have the H removed though so they can return to pick up more and allow the process to continue.
In animals - The H combines with the product of glycolysis, PA. This forms lactic acid.
In plants and some microorganisms (e.g, yeast) - the pyruvic acid has a CO2 removed to form ethanal. Then the ethanal accepts the H to become reduced and forms ethanol.
Both these processes will yield only the ATPs made in glycolysis, i.e. 2 molecules of ATP. Compared to aerobic respiration this is very poor.
However, that is not the only problem. Eventually these products of ethanol and lactic acid become toxic to the organisms.
The formation of ethanol cannot be reversed so if it gets to a high enough level it will kill the organism. The formation of lactic acid can be reversed but needs oxygen.
When oxygen becomes available to the animal again, it is converted into CO2 and water in aerobic respiration.
That is why anaerobic respiration cannot carry on indefinitely. Anaerobic respiration builds up an oxygen debt and that has to paid off and clearly the sooner, the better - e.g, when lifting weights in your arms, your muscles will start to burn. This is an effect of the lactic acid building up.