There is no real answer to your question because it is to vague. It is not clear what a "problem" is for you nor what "more posts" mean either.
When you need to search for something, the more elements you have, the more time it will take (that usually applies to most things in life). If you consider getting slower queries a problem, then I'm sorry to tell you that will happen. How slow they will run will depend on the amount of posts and the resources the hosting company has assigned.
Queries that take many seconds or even minutes might generate other issues such as timeouts (depending on server configuration) which usually result in an awful-looking error in the client side.
The last thing to consider is that there is a hard limit on the amount of posts you can have and it is 4294967295. I hardly believe someone has ever reached that number.
Just monitor your site and make sure no query takes more than 5 seconds. If they do, then there might be some issue either with your hosting company, a plugin or the Q2A core (in order of likeliness). And of course, you have already a much more serious issue with that "Temporary unavailability" error you're getting, which by order of likeliness should be a hosting company issue :)