What level of game can you afford to play?
The answer is very subject to the level of risk you personally would accept. There's also the questions of what happens if you do lose your entire bankroll due to bad luck. (Which happens!)
Generally accepted rules of thumb:
note: these guidelines are for solid players. If you are just a mediocre winner, these bankrolls will not be large enough. If you are a poor player, no bankroll can be safe.
In limit hold'em, you need about 300 Big Bets (600 Big Blinds) to have a reasonably safe bankroll. For a $2/$4 limit game, that means about $1200 dollars.
In no-limit hold'em, you need about 2000 Big Blinds to have a reasonably safe bankroll. For a $100 no-limit game with a $1 blind, you need about $2000 dollars.
In both examples, you would still enter a typical table with $100; you can lose your buy-in several times without getting tapped out and having to stop. You're relatively unlikely to lose so many times in a row that your entire bankroll is gone. The higher requirement for no-limit is a simple consequence of the higher variance of the game.
These limits are based on a strong player who wants a very low chance of losing their entire bankroll. If you can easily replace your bankroll, you can accept higher risk. For example, if you have a well-paid regular job, and you can afford to have $500 of "fun money" for poker, you can deposit it online and play 2/4 limit. Your chance of burning through all $500 on a bad run is much, much higher than if you had $1200 in there, but the loss of the $500 is not catastrophic, and you can re-deposit after a few more weeks of regular work.
On the other hand, if you don't have a lot of extra money, and you need to really safeguard your bankroll, you may want even higher bankrolls. A full-time player making a living at poker may want a bankroll higher still; after all, losing the bankroll means total loss of work!
When playing short-handed or heads-up poker, it is generally wise to have an even bigger bankroll than recommended here.
For short-handed (6max) limit holdem, a bankroll of around 500 Big Bets should be enough to survive the higher variance associated with these games, as 100 and 200 Big Bet down swings are much more common when playing with fewer opponents.
To play heads-up limit holdem, a bankroll nearer 1000 Big Bets is recommended, as you will most likely not have such a big advantage over your opponents (however well you play), meaning heads-up poker has the highest variance of all thus needing the biggest bankroll to fall back on.