If you have submitted your site again and again to Search Engines but you are still unable to get it indexed, it MAY be your Web hosting provider who is responsible.
If the domain sharing your server and IP Address is penalized by a Search Engine on account of spamming then your website is also expected to be banned or penalized. This will happen if there is virtual shared IP hosting.

Another situation may arise if your website is residing on a server containing illegal adult content, and that site is on the black list of Search Engines your website will be banned. While you should never assume that your site is blacklisted, you should also take precautions. Even if your site is dismissed from the rankings temporarily, you can probably clear up the mistake by contacting the Search Engine.

To be safe its best to host with a reputed Web Hosting Provider.
Your Web Hosting provider should be up 24/7. For proper website service it must be up 100% of the time so that Users and Search Engines aren't faced with a blank page or a 404 error. Search engines have no specific schedules for crawling, so your website must be up at all times in order to maintain your Search Engine Ranking.

Reputed Web Hosting Provider

The Reputation of the web host should be good with Search Engines.
Although there is no clear way to identify them, web hosts that are very popular could be trusted.
Some of the popular SEO Friendly Hosting providers are

Directi
Interland
1and1
Enom etc.

Sites hosted by free web hosting providers do not usually rank well in Search Engines for competitive keywords.

Conclusion

You may benefit from sticking with the most popular hosting service providers, even if they are a little more expensive, rather than going for the best deal of a host with no reputation.
It is also preferable to have an individual IP address so that risk of index removal is minimized (Usually Hosting providers provide dedicated IP Addresses only with purchase of SSL).