When a startup should hire a sys admin?

The startup is growing and you have now 5 full-time devs on your team. You have an older application which is up and running in production and a new web application almost ready to be released. You have something like 5 physical servers (some new, some outdated), and 5/10 "cloud machines". The old version of your application was running to a simple webserver and evolved overtime to something more "professional". On the other hand, your new application takes profit of the cloud environment and is designed to support "high traffic volume" by using queues, nosql and other cool stuffs... Your sales are increasing, features development and bug fixing are needed. Your devs are at 110% of their capacity (and become real "Software swiss army knife developers") by developing a new app and supporting the old one... As you run out of resources, when it comes to resolve specifics problems, you call "a guy who knows" by paying expensive bills. You have unix and windows servers and you install updates and read the log viewer when you have the time or when you get an application error. So far, an ordinary situation for a tech startup...

So, now you have the picture. Should this company hire a sys admin or let the "devs who want" time to operate the infrastructure and outsource when they have doubt about the solution ??

I say, as soon as you can afford one.

Devs aren't systems guys. They have no idea about how to best optimise MySQL to get the best performance. They don't know about Keepalive on Apache when running on a VM.
Those are 2 examples I've come across of a developer setting up a startup's webserver, and buggering it up in the process.

Many developers also have the attitude "It worked on my PC, therefore it must work on the server!" - Bad News.

Hire a systems guy if you want it done right.

When the cost of the staff outweighs the risk to your business of outages and data loss that would be prevented by the staff.

It's as simple as that.

At this scale you should already be looking at recording sufficient detail about what the current employees are spending their time on to answer the question. From your description I'd be getting everybody to track which product/site/service they are spending time on, what they are doing (admin, analysis, pre-launch development, bug fixing, other). If you've got near one FTE on the admin stuff then you certainly need an admin.

Beyond taking over some of the donkey work from your developers, a good admin should also add value to the services - particularly with regard to availability, security and performance.

But bear in mind this is a painful stage in the growth of your business - don't expect one person to be able to provide all the admin work 24x7 in perpetuity - some of the work must be shared so that other people have the skills (and access to backup media, passwords, external contacts....) to cover for her when she goes on vacation or gets a job elsewhere.