I have recently made a new website, built over 300+ links to it (which finally shown up on google webmaster tools btw), but it seems my site is in the sandbox. My site doesn't appear in the results even when I search for "domain.com". It used to appear though, so that's why I speculate this. My site is indexed so it's not that much of a problem. All I want to know is how long do sites remain in the sandbox. A couple of weeks, months? I'm questioning this because this a time sesntisve website, meaning that if it doesn't rank for the keyword(s) by a certain time, it will be worthless. Please don't tell me, "oh, it takes months to rank for keywords", because I researched my competition and keywords nicely, and have built over 300 links to my site for my keywords already.
Nobody knows as Google hasn't given any information about this and there has been no definitive tests to figure it out. Results have also varied from website to website. So there is no correct answer for this.
FYI, having 300 links to your site means absolutey nothing. First of all, you can't get links to your website. You get links to your website's pages. Search engines rank web pages, not websites. Second, quantity of links means nothing. Quality is what matters. I'd bet a large sum of money that those 300 links you got a low quality links so don't expect them to help you much, if anything, at all. Third, you sound like you expect your site to rank well instantly. I hate to be the one to break the news to you but it isn't going to happen. Uness you are target very long tail keywords it's going to take a while to rank well and that if you have quality content. If you don't it's not ever going to happen as you'll never get the necessary quality links you need to do it and the Panda agorithm will take its toll.
"Sandbox" is a rather nebulous SEO term but, in general, it means I think my site is good but Google doesn't rank it well. It's also used by slick SEO consultants as an excuse as to why a client's site isn't ranking without blaming their work, or lack thereof, as the cause. Basically, it's a meaningless term that doesn't convey anything beyond "I'm not happy with my search results placement in Google."
To get quick ranking for a site you'll need strong links from trusted authority sites. For example, I've seen a number of sites rank well within just a few weeks due to links from prominent news sites and blogs. More typical sites will require a lot more time depending on the targeted keyword(s), competition and so forth. It's really not possible to give a specific time frame since each site is different.
If your site isn't showing up at all for yourdomain.com and is showing up for site:domain.com then could be under some type of penalty (if it's not showing up for either, it's deindexed, which is another matter). This could be from excessive downtime, it being flagged as a parked domain or a thin affiliate site or something else. You can find out if this is the case by using Google Webmaster Tools.