Does the period of hosting impact website rankings on search engines? If so, what should be the minimum period one should host?
The period of hosting has nothing to do with the ranking on various search engines. It's the content and time for which your website remains online. As soon as your site goes offline and web crawling bots found it, the reindexing starts.
So, in real sense taking an yearly hosting or monthly payable hosting has no effect on ranking.
Period of hosting does not, according to my experiences or anything I've read, have any effect on SEO.
What you may be thinking of is the length of time a domain has been registered. Some people think that the length a domain has been registered has an effect. See https://www.quora.com/Search-Engine-Optimization-SEO/Does-domain-registration-length-impact-SEO for some discussion on the topic. Personally I've not seen this have any effect - or any effect it might have is dwarfed by the normal good-content practices.
how long the site has actualy been hosted i think has little to do with it, as many sites can just be hosted with sedo parking or an apache start page for years before there is a website present, if your referring to the age of the domain, ie how long ago the domain was registered matt cutts from google has an interesting done a small video on it here - http://www.youtube.com/watch?v=Y1_1NQWQJ2Q
[Assuming what you're actually asking about is the age of the domain and not hosting:]
Here's Matt Cutts: How does Google determine domain age, and is it important for ranking?.
The short version is that domain age doesn't get measured directly but more as a sort of function of actually becoming "part of" the web, ie. getting links. Also, it's not very important, so don't worry about it.
As far as minimum period, you're never going to get a number from Google. After buyding your domain, throw up a holding page with some basic information as soon as possible, and maybe get a few links to it if you can. By the time you're ready to launch, it'll have existed and been indexed for a few months, and that's good in itself.