There are four things you need to keep in mind when it comes to money.
(1) Expected Value (EV). This is the fraction of each bet that you expect to win. If you have positive EV, you expect to win money in the long run, if you have negative EV, you expect to lose money in the long run. If you play blackjack by mimicking the dealer, you will have an EV of about -0.09 (house edge of 9%), which is very bad. If you play basic strategy, you will have an EV of about -0.005 (house edge of 0.5%), which is the best game in the casino. If you count cards and bet properly, you can have positive EV's, generally somewhere between +0.005 and +0.02 (player edge of 0.5% to 2%).
(2) Standard Deviation (SD). This is a measure of how far your actual results will vary from your EV. The larger your SD, the more "luck" factors in. If you play basic strategy, your SD will be around 1.15, which means that for each hand you play, you can expect to be between EV-SD = -1.155 and EV+SD = +1.145 about 84% of the time, and between EV-2*SD = -2.305 and EV+2*SD = +2.295 about 95% of the time.
(3) Bankroll. This is the amount of money you've set aside to gamble with. It should be separate from anything that you need to live on, both from the perspective that you shouldn't be gambling with rent money and from the perspective that you need to keep good records to evaluate yourself. Bankrolls range anywhere from a few hundred dollars for recreational players playing low stakes to millions of dollars for seasoned pros playing high stakes.
(4) Risk of Ruin (ROR). This is the likelihood that your bankroll will hit 0 at some point and you will be "broke." Obviously, for professionals, ROR needs to be very small because they have a lot to lose (<<1%), whereas for recreational players a higher ROR (5-10% and maybe even higher) is acceptable.
The most important thing to realize is that all four of these things are interconnected. Generally, if you know three of these things, you can calculate the fourth; if you know zero of these things, it's pretty difficult to estimate any of them.
Your bet spread and playing style will dictate your EV and SD. There are computer programs to do those calculations for you. Then you need to either decide how much of a bankroll you have to calculate your ROR, or you need to decide how much of a ROR you're willing to stomach and calculate how much of a bankroll you'll need. Again, there are programs which will do the math for you.
As a starting point, $250-$300 isn't bad if you can save more. Just realize that it's basically one shot, if you have bad luck you'll go busto really quickly. I'd say try it once you're confident of your skills - who knows, you might realize you hate blackjack and it's not fun and you don't want to continue.
If you plan to play regularly, you'll need more. If you eventually want to go pro, you'll need even more than that. There's no good way to answer your question without knowing more specifics.
