Can you pay life as an additional cost if you don't have enough life to survive?

Paying life is a cost, and costs are paid during the casting of a spell or activation of an ability. There is no rule that prevents you from paying life you don't 'have enough of' to survive. You simply pay the required amount, reducing your life total accordingly, even into negative numbers.

State-based actions — specifically CR 704.5a — check whether a player has 0 or less life, but these checks only happen at specific times: after a spell or ability finishes resolving, or when a player would receive priority. They do not interrupt the process of paying costs mid-cast (CR 117.3).

This means you can legally cast a spell with an additional life payment of, say, 4 life when you only have 2 life remaining. Your life total drops to −2. Once the spell resolves (or even before it resolves, after you finish casting it and state-based actions are checked), you will lose the game due to having 0 or less life per CR 704.5a.

Concrete example: You have 2 life and cast Hatred, paying its additional cost of 4 life. Your life total becomes −2. The spell is now on the stack. Before it resolves, state-based actions are checked and you lose the game at −2 life — but the cost payment itself was perfectly legal.

Note: if something replaces your loss (e.g., you have a Platinum Angel in play), you simply continue playing at a negative life total. The cost payment is always valid regardless.